spectral functions, the geometric power of eigenvalues,
TRANSCRIPT
![Page 1: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/1.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral FunctionsThe Geometric Power of Eigenvalues
Pedro Fernando Morales
Department of MathematicsBaylor University
pedro moralesbayloredu
Athens Ohio 10182012
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 2: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/2.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 3: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/3.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 4: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/4.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 5: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/5.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 6: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/6.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 7: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/7.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Outline
1 Introduction
2 Laplace-type Operators
3 Heat kernel
4 Zeta function
5 Regularization
6 Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 8: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/8.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 9: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/9.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 10: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/10.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 11: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/11.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 12: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/12.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 13: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/13.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 14: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/14.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 15: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/15.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 16: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/16.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 17: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/17.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 18: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/18.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 19: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/19.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ
(eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 20: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/20.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Eigenvalues (a little more formal)
bull Point spectrum of an operator
bull P minus λI not bounded below (not injective)
bull Pφ = λφ (eigenvalue equation)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 21: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/21.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 22: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/22.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 23: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/23.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 24: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/24.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 25: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/25.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Philosophical question
What is an eigenvalue
bull A way to linearize a problem
bull Measures the distortion of a system
bull Decomposes an object into simpler pieces
bull Determines the resolution of a method
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 26: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/26.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 27: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/27.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 28: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/28.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 29: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/29.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Other names and similar ideas
bull Fourier expansion
bull Characters of representations
bull Decomposition into irreducibles
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 30: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/30.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 31: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/31.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 32: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/32.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 33: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/33.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 34: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/34.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 35: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/35.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Differential Operators
bull M a compact manifold
bull E a vector bundle over M
bull P Γ(E )rarr Γ(E ) a differential operator over M
bull With boundary conditions if partM 6= empty
Eigenvalue Equation
Pφ = λφ
where φ isin Γ(E )
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 36: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/36.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
(Vibrating membrane)
Pedro Fernando Morales Math Department
Spectral Functions
![Page 37: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/37.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifold
E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M
V isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )
nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 38: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/38.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifold
E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M
V isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )
nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 39: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/39.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
There is a close relation between the shape of the manifold andthe eigenvalues (eigenfunctions) of the Laplacian
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifold
E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M
V isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )
nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 40: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/40.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifold
E a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M
V isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )
nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 41: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/41.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over M
V isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )
nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 42: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/42.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )
nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 43: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/43.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on E
g metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 44: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/44.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 45: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/45.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Generalized Laplace equation
M d-dimensional smooth compact Riemannian manifoldE a smooth vector bundle over MV isin End(E )nablaE a connection on Eg metric on M
Laplace-type
P Γ(E )rarr Γ(E ) is a Laplace-type differential operator if P can bewritten as
P = minusg ijnablaEi nablaE
j + V
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 46: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/46.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric
(Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 47: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/47.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)
Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 48: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/48.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvalues
Bounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 49: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/49.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded below
Tend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 50: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/50.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties Laplace-type Operators
Symmetric (Self-adjoint with suitable boundary conditions)Real eigenvaluesBounded belowTend to infinity
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 51: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/51.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 52: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/52.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 53: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/53.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 54: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/54.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 55: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/55.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 56: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/56.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Recall
Heat Equation
∆u = ut
for a domain D where u = 0 at partD and u|t=0 = f (x) is the initialheat distribution
we use the heat kernel to find the heat distribution at any time
bull Kt(t x y) = ∆K (t x y) for x y isin D t gt 0
bull limtrarr0
K (t x y) = δ(x minus y)
bull K (t x y) = 0 for x or y in partD
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 57: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/57.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 58: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/58.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Solving the heat equation
(Tf )(t x) =
intDK (t x y)f (y)dy
solves the heat equation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 59: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/59.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 60: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/60.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 61: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/61.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 62: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/62.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat kernel for a Laplace-type operator
Likewise for a Laplace-type operator
Heat Kernel
bull (partt minus P)K (t x y) = 0 Cinfin (R+MM)
bull limtrarr0
intMK (t x y)f (y) = f (x)forallf isin L2(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 63: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/63.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 64: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/64.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 65: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/65.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 66: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/66.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Relation with eigenvalues
K (t x y) = etP
=sumλ
eminustλφλ(x)φλ(y)
where λ runs over the eigenvalues of P and φλ is thecorresponding eigenfunction
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 67: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/67.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 68: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/68.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 69: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/69.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 70: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/70.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Heat Kernel Asymptotic Expansion
For small values of t
Asymptotic Expansion
Kt(t x x) simsum
k=0121
bk(x)tkminusd2
Heat kernel coefficients
ak =
intMbk(x)dx
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 71: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/71.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 72: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/72.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 73: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/73.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 74: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/74.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 75: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/75.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Properties
bull Provide geometric information about the manifold
bull a0 is the volume of M
bull a12 is the volume of the boundary partM
bull ak has curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 76: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/76.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 77: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/77.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)
Recall K (t) =sum
λ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 78: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/78.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 79: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/79.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 80: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/80.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 81: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/81.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 82: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/82.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Spectral Functions
The heat kernel is an example of an spectral function (defined interms of the spectrum of P)Recall K (t) =
sumλ eminustλ
Zeta function
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 83: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/83.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 84: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/84.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 85: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/85.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta function
Problem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 86: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/86.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem
only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 87: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/87.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2
All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 88: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/88.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Zeta function
Zeta function
The zeta function associated with the operator P is defined by
ζP(s) =sumλ
λminuss
eg λn = n gives the Riemann zeta functionProblem only defined for lt(s) gt d2All the important information lies to the left of this region
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 89: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/89.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 90: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/90.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expression
Donrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 91: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/91.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergence
Rather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 92: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/92.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Regularization
Is a method of making sense of a divergent expressionDonrsquot take the usual meaning of convergenceRather look at the meaning of the sum
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 93: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/93.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 94: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/94.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot =
minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 95: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/95.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 96: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/96.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn
=1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 97: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/97.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 98: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/98.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n
=1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 99: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/99.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 100: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/100.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1
We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 101: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/101.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Example
1 + 2 + 4 + 8 + 16 + middot middot middot = minus 1
Special case ofinfinsumn=0
rn =1
1minus r
infinsumn=0
2n =1
1minus 2= minus1
Convergent only for |r | lt 1We just made an analytic continuation
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 102: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/102.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 103: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/103.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 104: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/104.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Analytic continuation
ζP(s) admits an analytic continuation to the whole complex planeexcept for simple poles at s = d2 (d minus 1)2 12minus(2n+ 1)2for n a non-negative integer
Residues
Res ζP (s) =ad2minussΓ(s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 105: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/105.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 106: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/106.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 107: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/107.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)
ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 108: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/108.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)
Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 109: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/109.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zero
No curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 110: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/110.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 111: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/111.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Riemann Zeta
bull Only one pole (simple) at s = 1
bull Res ζR(1) = 1
a0 = Γ(d2)ak = 0 (No other residues)Volume of the boundary is zeroNo curvature terms
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 112: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/112.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifold
d = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 113: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/113.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1
length=radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 114: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/114.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 115: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/115.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 116: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/116.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 117: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/117.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditions
eigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 118: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/118.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 119: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/119.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Manifoldd = 1length=
radicπ
Operator
minus d2
dx2φ = λφ
Dirichlet boundary conditionseigenvalues n2 n isin N
ζP(s) = ζR(2s)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 120: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/120.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behavior
The geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 121: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/121.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 122: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/122.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Meromorphic structure
Convergence problems(poles) come from the large λ behaviorThe geometric information is encoded in the asymptotic behaviorof the eigenvalues
Weylrsquos law
Let N(λ) be the number of eigenvalues less than λ then
N(λ) sim 1
(4π)d2Γ(d2)Vol(M)λd2
where d = dim(M)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 123: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/123.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized Trace
Functional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 124: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/124.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional Determinant
Spectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 125: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/125.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimension
PhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 126: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/126.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysics
Casimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 127: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/127.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 128: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/128.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)
Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin N
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 129: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/129.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Applications
Zeta Regularized TraceFunctional DeterminantSpectral dimensionPhysicsCasimir Energy λn eigenvalues of the Hamiltonian
ECas =infinsumn=1
radicλn
One-Loop Effective Action (Functional Determinant)Heat Kernel Coefficients
ad2minusz = Γ(z) Res ζP(z)
for z = d2 (d minus 1)2 12minus(2n + 1)2 n isin NPedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 130: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/130.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Casimir Effect
Is a quantum field effect that arises when considering vacuumfluctuations
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 131: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/131.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 132: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/132.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 133: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/133.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Why is so important
bull Believed to explain the stability of an electron
bull Very sensitive to the geometry of the space (Quantum andComsmological implications)
bull Provides a better understanding of the zero-point energy
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 134: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/134.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 135: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/135.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 136: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/136.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 137: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/137.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Conclusions
bull Eigenvalues know a lot of the geometry of a system
bull Describe the dynamics in a geometric object
bull New information appear when regularizing expressions
bull Useful to describe high energy systems (quantum physics)
Pedro Fernando Morales Math Department
Spectral Functions
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-
![Page 138: Spectral Functions, The Geometric Power of Eigenvalues,](https://reader033.vdocuments.us/reader033/viewer/2022052601/55972c771a28ab49708b4824/html5/thumbnails/138.jpg)
Introduction Laplace-type Operators Heat kernel Zeta function Regularization Applications
Questions
Thank you
Pedro Fernando Morales Math Department
Spectral Functions
- Introduction
- Laplace-type Operators
- Heat kernel
- Zeta function
- Regularization
- Applications
-