etudes in spectral theory - victor...

Etudes in Spectral Theory

Math Physics Seminar at Tel-Aviv University

Victor Ivrii

Department of Mathematics, University of Toronto

November 7, 2019

Victor Ivrii (Math., Toronto) Etudes in Spectral Theory November 7, 2019 1 / 48

Table of Contents

Table of Contents

1 Distribution of EigenvaluesVariational MethodsTauberian Methods

2 Equidistribution of eigenfunctions

3 Can one hear the shape of the drum?

4 Nodal lines

5 Exotic spectraMixed spectraBand spectraLandau levelsTen martini problem


Introduction

Introduction

I briefly describe five old but still actively explored problems of theSpectral Theory of Partial Differential Equations

1 How eigenvalues are distributed (where eigenvalues often meansquares of the frequencies in the mechanical or electromagneticproblems or energy levels in the quantum mechanics models) and therelation to the behaviour of the billiard trajectories.

2 Equidistribution of eigenfunctions and connection to ergodicity ofbilliard trajectories (a quantum quantum ergodicity and a classicalquantum ergodicity).

3 Can one hear the shape of the drum?

4 Nodal lines and Chladni plates.

5 Strange spectra of quantum systems.


Reminder

Reminder

Spectrum of operator H (Spec(H)) we call the set of all 𝜆 ∈ C, for whichresolvent (H − 𝜆)−1 either does not exist or is an unbounded operator.

For matrices (operators in the finite-dimensional space) spectrum is a setof eigenvalues, such 𝜆, that (H − 𝜆)u = 0 for some u = 0 (which is calledan eigenvector). For operators (in the infinite dimensional space) the set ofall eigenvalues is a point spectrum. There are other types of the spectrum.There are two classifications.


Reminder

Reminder

Spectrum of operator H (Spec(H)) we call the set of all 𝜆 ∈ C, for whichresolvent (H − 𝜆)−1 either does not exist or is an unbounded operator.For matrices (operators in the finite-dimensional space) spectrum is a setof eigenvalues, such 𝜆, that (H − 𝜆)u = 0 for some u = 0 (which is calledan eigenvector).

For operators (in the infinite dimensional space) the set ofall eigenvalues is a point spectrum. There are other types of the spectrum.There are two classifications.


Reminder

Reminder

Spectrum of operator H (Spec(H)) we call the set of all 𝜆 ∈ C, for whichresolvent (H − 𝜆)−1 either does not exist or is an unbounded operator.For matrices (operators in the finite-dimensional space) spectrum is a setof eigenvalues, such 𝜆, that (H − 𝜆)u = 0 for some u = 0 (which is calledan eigenvector). For operators (in the infinite dimensional space) the set ofall eigenvalues is a point spectrum. There are other types of the spectrum.

There are two classifications.


Reminder

Reminder

Spectrum of operator H (Spec(H)) we call the set of all 𝜆 ∈ C, for whichresolvent (H − 𝜆)−1 either does not exist or is an unbounded operator.For matrices (operators in the finite-dimensional space) spectrum is a setof eigenvalues, such 𝜆, that (H − 𝜆)u = 0 for some u = 0 (which is calledan eigenvector). For operators (in the infinite dimensional space) the set ofall eigenvalues is a point spectrum. There are other types of the spectrum.There are two classifications.


Reminder

General classification

For operators in the Banach space the are

1 Point spectrum—see above.

2 Continuous spectrum: 𝜆 is not an eigenvalue, the range of (H − 𝜆) isdense, but (H − 𝜆)−1 is an unbounded operator.

3 Residue spectrum—the rest: i.e. 𝜆 is not an eigenvalue, and therange of (H − 𝜆) is not dense.

However we are interested in the different classification–for self-adjointoperators in the Hilbert space. Point spectrum is defined in the same way,residue spectrum is empty, but continuous spectrum is defined differentlyand there are two types of it.


Reminder






However we are interested in the different classification–for self-adjointoperators in the Hilbert space.

Point spectrum is defined in the same way,residue spectrum is empty, but continuous spectrum is defined differentlyand there are two types of it.


Reminder






However we are interested in the different classification–for self-adjointoperators in the Hilbert space. Point spectrum is defined in the same way,residue spectrum is empty, but continuous spectrum is defined differentlyand there are two types of it.


Reminder

Classification for self-adjoint operators

For self-adjoint operators in the Hilbert space there is a spectraldecomposition: a family of orthogonal spectral projectors E (𝜆), s.t.E (𝜆)E (𝜆′) = E (𝜆′) for 𝜆′ < 𝜆,

E (−∞) = 0, E (∞) = I , E (𝜆) issemi-continuous from the left, i.e. E (𝜆−) = E (𝜆), and∫

𝜆d𝜆E (𝜆) = H.


Reminder

Classification for self-adjoint operators

For self-adjoint operators in the Hilbert space there is a spectraldecomposition: a family of orthogonal spectral projectors E (𝜆), s.t.E (𝜆)E (𝜆′) = E (𝜆′) for 𝜆′ < 𝜆, E (−∞) = 0, E (∞) = I , E (𝜆) issemi-continuous from the left, i.e. E (𝜆−) = E (𝜆), and∫

𝜆d𝜆E (𝜆) = H.


Reminder

Furthermore, Hilbert space H is decomposed into a direct sumH = Hpp ⊕ Hac ⊕ Hsc, s.t. projectors E (𝜆) maps each of them into itself,then operator H is decomposed into a direct sum H = Hp ⊕Hac ⊕Hsc,

andrestriction of E (𝜆) to Hpp, Hac and Hsc is an atomic measure, absolutelycontionuos measure and singular continuous measure respectively.

Then the spectra of operators Hac and Hsc is called absolutely continuousspectrum and singular continuous spectrum of operator H respectivelyContinuous spectrumis an union of the absolutely continuous and singularcontinuous spectra.

And the spectrum of Hpp is a closure of the set of eigenvalues (which isnot necessarily closed). We call it (not universally accepted) purely pointspectrum.


Reminder

Furthermore, Hilbert space H is decomposed into a direct sumH = Hpp ⊕ Hac ⊕ Hsc, s.t. projectors E (𝜆) maps each of them into itself,then operator H is decomposed into a direct sum H = Hp ⊕Hac ⊕Hsc, andrestriction of E (𝜆) to Hpp, Hac and Hsc is an atomic measure, absolutelycontionuos measure and singular continuous measure respectively.

Then the spectra of operators Hac and Hsc is called absolutely continuousspectrum and singular continuous spectrum of operator H respectivelyContinuous spectrumis an union of the absolutely continuous and singularcontinuous spectra.

And the spectrum of Hpp is a closure of the set of eigenvalues (which isnot necessarily closed). We call it (not universally accepted) purely pointspectrum.


Reminder

So, each spectrum is a closed set and the different spectra may overlap.

The definition of the multiplicity of the eigenvalue is wide-known (thedimension of the corresponding eigenspace) and one can define themultiplicity of the absolutely continuous spectrum.Discrete spoectrum of operator H is a set of its eigenvalues of the finitemultiplicity, which are isolated from the rest of the spectrum.The remaining spectrum (eigenvalues of the infinite multiplicity,accumulation points of the point spectrum, points of the continuousspectrum) make an essential spectrum of operator H.


Reminder


The definition of the multiplicity of the eigenvalue is wide-known (thedimension of the corresponding eigenspace) and one can define themultiplicity of the absolutely continuous spectrum.

Discrete spoectrum of operator H is a set of its eigenvalues of the finitemultiplicity, which are isolated from the rest of the spectrum.The remaining spectrum (eigenvalues of the infinite multiplicity,accumulation points of the point spectrum, points of the continuousspectrum) make an essential spectrum of operator H.


Reminder


The definition of the multiplicity of the eigenvalue is wide-known (thedimension of the corresponding eigenspace) and one can define themultiplicity of the absolutely continuous spectrum.Discrete spoectrum of operator H is a set of its eigenvalues of the finitemultiplicity, which are isolated from the rest of the spectrum.

The remaining spectrum (eigenvalues of the infinite multiplicity,accumulation points of the point spectrum, points of the continuousspectrum) make an essential spectrum of operator H.


Reminder


The definition of the multiplicity of the eigenvalue is wide-known (thedimension of the corresponding eigenspace) and one can define themultiplicity of the absolutely continuous spectrum.Discrete spoectrum of operator H is a set of its eigenvalues of the finitemultiplicity, which are isolated from the rest of the spectrum.The remaining spectrum (eigenvalues of the infinite multiplicity,accumulation points of the point spectrum, points of the continuousspectrum) make an essential spectrum of operator H.


Distribution of Eigenvalues Variational Methods

Distribution of Eigenvalues

In 1911 a 26-years old mathematician, a former studentof David Hilbert, Hermann Weyl published a very impor-tant paper Uber die asymptotische Verteilung der Eigen-werte (About the asymptotic distribution of eigenvalues),which followed by four more papers in 1912, 1913 and1915.

Hermann Weyl

In this paper he proved a formula, later called Weyl’s Law, which wasconjectured independently by Arnold Sommerfeld and Hendrik Lorentz in1910.

According to the book of Lord Rayleigh The Theory of Sound (1887)eigenvalues were the squares of the frequencies; in the Quantum Mechanics(developed a bit later) eigenvalues were primarily the energy levels.


https://en.wikipedia.org/wiki/Arnold_Sommerfeld

https://en.wikipedia.org/wiki/Hendrik_Lorentz

https://en.wikipedia.org/wiki/John_William_Strutt,_3rd_Baron_Rayleigh


For Dirichlet Laplacian in a bounded d-dimensional domain Ω H.Weyl(1911) proved that

N(𝜆) = c0mes(Ω)𝜆d2 + o(𝜆

d2 ) (1)

as 𝜆→ +∞ and

in 1913 conjectured that

N(𝜆) = c0mes(Ω)𝜆d2 ∓ c1mesd−1(𝜕Ω)𝜆

d−12 + o(𝜆

d−12 ). (2)

Definition 1

Here N(𝜆) is a number of eigenvalues of Laplacian −Δ, which are lessthan 𝜆.

In this framework eigenvalue is the non-trivial solution to the equation

−Δ𝜓n := −(𝜕2x1 + 𝜕2x2 + . . .+ 𝜕2xd

)𝜓n = 𝜆n𝜓n, (3)

satisfying the boundary conditions (Dirichlet or Neumann).



For Dirichlet Laplacian in a bounded d-dimensional domain Ω H.Weyl(1911) proved that

N(𝜆) = c0mes(Ω)𝜆d2 + o(𝜆

d2 ) (1)

as 𝜆→ +∞ and in 1913 conjectured that

N(𝜆) = c0mes(Ω)𝜆d2 ∓ c1mesd−1(𝜕Ω)𝜆

d−12 + o(𝜆

d−12 ). (2)

Definition 1

Here N(𝜆) is a number of eigenvalues of Laplacian −Δ, which are lessthan 𝜆.

In this framework eigenvalue is the non-trivial solution to the equation

−Δ𝜓n := −(𝜕2x1 + 𝜕2x2 + . . .+ 𝜕2xd

)𝜓n = 𝜆n𝜓n, (3)

satisfying the boundary conditions (Dirichlet or Neumann).



To explain (1) consider a rectangular box of the size a1 × a2 (for asimplicity we take d = 2). Then the eigenvalue problem could be solvedeasily by the separation of the variables

and N(𝜆) equals to the number ofinteger points in the domain

Θ ={(m1,m2) ∈ Z+2,

m21

a21+

m22

a22<

𝜆

𝜋2}, (4)

which should be approximately equal to the volume of the domain Θ

(which is 1/4 of the ellipse with semiaxis 𝜋−1𝜆12 a1, 𝜋

−1𝜆12 a2) i.e.

𝜔2(2𝜋)−1𝜆

12 a1 × (2𝜋)−1𝜆

12 a2 =

𝜔2(2𝜋)−2𝜆

d2 mes(Ω),

mes(Ω) = a1a2,

which would be exactly the first term in (1). Here and below 𝜔d is avolume of the unit ball in Rd .



To explain (1) consider a rectangular box of the size a1 × a2 (for asimplicity we take d = 2). Then the eigenvalue problem could be solvedeasily by the separation of the variables and N(𝜆) equals to the number ofinteger points in the domain

Θ ={(m1,m2) ∈ Z+2,

m21

a21+

m22

a22<

𝜆

𝜋2}, (4)

which should be approximately equal to the volume of the domain Θ

(which is 1/4 of the ellipse with semiaxis 𝜋−1𝜆12 a1, 𝜋

−1𝜆12 a2) i.e.

𝜔2(2𝜋)−1𝜆

12 a1 × (2𝜋)−1𝜆

12 a2 =

𝜔2(2𝜋)−2𝜆

d2 mes(Ω),

mes(Ω) = a1a2,

which would be exactly the first term in (1). Here and below 𝜔d is avolume of the unit ball in Rd .



Weyl conjecture (2) was the result of a more precise analysis of the sameproblem: for Dirichlet boundary condition we do not count points withmi = 0 (blue) and for Neumann problem we count them, so in (2) will be“−” for Dirichlet, and “+” for Neumann, c1 =

14(2𝜋)

1−d𝜔d−1.



The proof of (1) by Weyl was based on this formula for boxes, integralequations and variational arguments he invented. Those arguments arebased on the formula

N(𝜆) = max dimL, (5)

where L runs over all subspaces of H on which quadratic form‖∇u‖2 − 𝜆‖u‖2 is negative and H is Sobolev space H1(Ω) in the case ofNeumann boundary problem and H1

0 (Ω) = {u ∈ H1(Ω), u|𝜕Ω = 0} in thecase of Dirichlet boundary problem.

Using this formula, corresponding expression for boxesand a partition of the domain Ω into boxes Weyl proved(1).

Richard Courant in 1920, pushing this method to its

limits proved remainder estimate to O(𝜆d−12 log 𝜆) for

bounded domains with C∞ boundary.Actually both H. Weyl and R. Courant considered onlyd = 2, 3.

Richard Courant






0 (Ω) = {u ∈ H1(Ω), u|𝜕Ω = 0} in thecase of Dirichlet boundary problem.Using this formula, corresponding expression for boxesand a partition of the domain Ω into boxes Weyl proved(1).

Richard Courant in 1920, pushing this method to its



Richard Courant






0 (Ω) = {u ∈ H1(Ω), u|𝜕Ω = 0} in thecase of Dirichlet boundary problem.Using this formula, corresponding expression for boxesand a partition of the domain Ω into boxes Weyl proved(1).Richard Courant in 1920, pushing this method to its



Richard Courant



In sixties and after this vari-ational methods were per-fected by Michael Birman andMichael Solomyak (workingmainly together), and by manyof their students (most no-tably by Grigori Rozenblioum)

and by Elliott Lieb, and BarrySimon and by their students.

In these papers much moregeneral problems were consid-ered and much more subtle ar-guments were applied.However these methods werenot sufficient to prove evenO(𝜆(d−1)/2) remainder esti-mate, leave alone Weyl’s con-jecture.

Michael Michael Grigoryi

Birman Solomyak Rozenblioum

Elliott Lieb Barry Simon





In these papers much moregeneral problems were consid-ered and much more subtle ar-guments were applied.

However these methods werenot sufficient to prove evenO(𝜆(d−1)/2) remainder esti-mate, leave alone Weyl’s con-jecture.








In these papers much moregeneral problems were consid-ered and much more subtle ar-guments were applied.However these methods werenot sufficient to prove evenO(𝜆(d−1)/2) remainder esti-mate, leave alone Weyl’s con-jecture.





Distribution of Eigenvalues Tauberian Methods

Tauberian Methods

In 1935 and 1936 Torsten Carleman invented very differ-ent approach.

Let us consider some function of the operator and a pa-rameter f (−Δ, t) such that its Schwartz kernel (i.e. in-tegral kernel) u(x , y , t) could be constracted by methodsof partial differential equations.Then the trace of f (−Δ, t) could be expressed as

Tr(f (−Δ, t)

)=

∫u(x , x , t) dx . (6)

TorstenCarleman

On the other hand, it is related to N(𝜆) by

Tr(f (−Δ, t)

)=

∫f (𝜆, t) d𝜆N(𝜆). (7)



Tauberian Methods

In 1935 and 1936 Torsten Carleman invented very differ-ent approach.Let us consider some function of the operator and a pa-rameter f (−Δ, t) such that its Schwartz kernel (i.e. in-tegral kernel) u(x , y , t) could be constracted by methodsof partial differential equations.

Then the trace of f (−Δ, t) could be expressed as

Tr(f (−Δ, t)

)=

∫u(x , x , t) dx . (6)

TorstenCarleman


Tr(f (−Δ, t)

)=

∫f (𝜆, t) d𝜆N(𝜆). (7)



Tauberian Methods

In 1935 and 1936 Torsten Carleman invented very differ-ent approach.Let us consider some function of the operator and a pa-rameter f (−Δ, t) such that its Schwartz kernel (i.e. in-tegral kernel) u(x , y , t) could be constracted by methodsof partial differential equations.Then the trace of f (−Δ, t) could be expressed as

Tr(f (−Δ, t)

)=

∫u(x , x , t) dx . (6) Torsten

Carleman


Tr(f (−Δ, t)

)=

∫f (𝜆, t) d𝜆N(𝜆). (7)



Then, calculating this trace by (6) we know the integral transform ofN(𝜆),

and knowing that N(𝜆) is a monotone non-decreasing function of 𝜆,allows us to recover N(𝜆) with some error.Since such theorems, helping us to partially recover a function from itsintegral transform are called Tauberian theorems, the method, invented byCarleman is called Tauberian method.Actually, there are different Tauberian methods, depending on the choiceof the function f .One of the popular choices is f (−Δ, t) = etΔ; then u is a fundamentalsolution to heat equation, i.e.

ut = Δxu, (8)

u|t=0 = δ(x − y). (9)

So, we have heat equation method. Such function u(x , y , t) is easy toconstruct but it is very difficult to prove a good remainder estimate fromLaplace transform

𝜎(t) :=

∫e−𝜆td𝜆N(𝜆). (10)



Then, calculating this trace by (6) we know the integral transform ofN(𝜆), and knowing that N(𝜆) is a monotone non-decreasing function of 𝜆,allows us to recover N(𝜆) with some error.

Since such theorems, helping us to partially recover a function from itsintegral transform are called Tauberian theorems, the method, invented byCarleman is called Tauberian method.Actually, there are different Tauberian methods, depending on the choiceof the function f .One of the popular choices is f (−Δ, t) = etΔ; then u is a fundamentalsolution to heat equation, i.e.

ut = Δxu, (8)

u|t=0 = δ(x − y). (9)


𝜎(t) :=




Then, calculating this trace by (6) we know the integral transform ofN(𝜆), and knowing that N(𝜆) is a monotone non-decreasing function of 𝜆,allows us to recover N(𝜆) with some error.Since such theorems, helping us to partially recover a function from itsintegral transform are called Tauberian theorems, the method, invented byCarleman is called Tauberian method.

Actually, there are different Tauberian methods, depending on the choiceof the function f .One of the popular choices is f (−Δ, t) = etΔ; then u is a fundamentalsolution to heat equation, i.e.

ut = Δxu, (8)

u|t=0 = δ(x − y). (9)


𝜎(t) :=




Then, calculating this trace by (6) we know the integral transform ofN(𝜆), and knowing that N(𝜆) is a monotone non-decreasing function of 𝜆,allows us to recover N(𝜆) with some error.Since such theorems, helping us to partially recover a function from itsintegral transform are called Tauberian theorems, the method, invented byCarleman is called Tauberian method.Actually, there are different Tauberian methods, depending on the choiceof the function f .

One of the popular choices is f (−Δ, t) = etΔ; then u is a fundamentalsolution to heat equation, i.e.

ut = Δxu, (8)

u|t=0 = δ(x − y). (9)


𝜎(t) :=




Then, calculating this trace by (6) we know the integral transform ofN(𝜆), and knowing that N(𝜆) is a monotone non-decreasing function of 𝜆,allows us to recover N(𝜆) with some error.Since such theorems, helping us to partially recover a function from itsintegral transform are called Tauberian theorems, the method, invented byCarleman is called Tauberian method.Actually, there are different Tauberian methods, depending on the choiceof the function f .One of the popular choices is f (−Δ, t) = etΔ; then u is a fundamentalsolution to heat equation, i.e.

ut = Δxu, (8)

u|t=0 = δ(x − y). (9)


𝜎(t) :=





ut = Δxu, (8)

u|t=0 = δ(x − y). (9)

So, we have heat equation method.

Such function u(x , y , t) is easy toconstruct but it is very difficult to prove a good remainder estimate fromLaplace transform

𝜎(t) :=





ut = Δxu, (8)

u|t=0 = δ(x − y). (9)

So, we have heat equation method. Such function u(x , y , t) is easy toconstruct

but it is very difficult to prove a good remainder estimate fromLaplace transform

𝜎(t) :=





ut = Δxu, (8)

u|t=0 = δ(x − y). (9)


𝜎(t) :=




In 1952 Boris Levitan invented hyperbolic operatormethod. He considered f (−Δ, t) = cos(t

√−Δ).

Then u(x , y , t) is a fundamental solution to wave equa-tion

utt = Δxu, (11)

u|t=0 = δ(x − y), ut |t=0 = 0. (12)

and we need to recover N(𝜆) from the (cos)-Fourier trans-form

𝜎(t) :=

∫cos(𝜆t)d𝜆N(𝜆

2). (13)

Boris Levitan

In this case the recovery part is easy, and one can get a good remainderestimate, but the PDE part is difficult near the border or for large t

.

Using construction for |t| ≤ T with some constant > 0 Levitan recoveredremainder estimate O(𝜆(d−1)/2) for compact closed manifolds (closed =without the boundary.




√−Δ).


utt = Δxu, (11)

u|t=0 = δ(x − y), ut |t=0 = 0. (12)


𝜎(t) :=


2). (13)Boris Levitan

In this case the recovery part is easy, and one can get a good remainderestimate, but the PDE part is difficult near the border or for large t

.





√−Δ).


utt = Δxu, (11)

u|t=0 = δ(x − y), ut |t=0 = 0. (12)


𝜎(t) :=



In this case the recovery part is easy, and one can get a good remainderestimate, but the PDE part is difficult near the border or for large t.





√−Δ).


utt = Δxu, (11)

u|t=0 = δ(x − y), ut |t=0 = 0. (12)


𝜎(t) :=



In this case the recovery part is easy, and one can get a good remainderestimate, but the PDE part is difficult near the border or for large t.Using construction for |t| ≤ T with some constant > 0 Levitan recoveredremainder estimate O(𝜆(d−1)/2) for compact closed manifolds (closed =without the boundary.



In 1968 Lars Hormander, using hyperbolic operatormethod and the new technique of Microlocal Analysis,recovered remainder estimate O(𝜆(d−1)/2) for more gen-eral operators on compact closed manifolds.

Lars Hormander

However estimate o(𝜆(d−1)/2) may fail without some additional condition.Indeed, consider Laplace operator on the sphere Sd , for d = 2. Then

eigenvalues are 𝜆m = m(m + 1)/2 of multiplicity 2m + 1 ≍ 𝜆1/2m and we

cannot get a remainder estimate better than O(𝜆(d−1)/2), and the same istrue for any dimension.




Lars Hormander

However estimate o(𝜆(d−1)/2) may fail without some additional condition.

Indeed, consider Laplace operator on the sphere Sd , for d = 2. Then






Lars Hormander

However estimate o(𝜆(d−1)/2) may fail without some additional condition.Indeed, consider Laplace operator on the sphere Sd , for d = 2. Then





In 1975 J. J. Duistermaat and Victor Guil-lemin, using hyperbolic operator method andMicrolocal Analysis recovered remainder es-timate o(𝜆(d−1)/2) for operators on compactclosed manifolds, under geometric condition,preventing high degeneration of eigenvalues. J.J. Victor

Duistermaat Guillemin

This condition, for the Laplace operator was

Non-periodicity condition

The measure of the set all periodic geodesics is zero.

Geodesics could be marked by their initial points and directions, which areelements of the phase space, and there is a standard volume measure onthe phase space.On the sphere all geodesics (large circles) are periodic, but there are othermanifolds (f.e. Zoll-Tanner manifolds) with the same property.








Geodesics could be marked by their initial points and directions, which areelements of the phase space, and there is a standard volume measure onthe phase space.

On the sphere all geodesics (large circles) are periodic, but there are othermanifolds (f.e. Zoll-Tanner manifolds) with the same property.








Geodesics could be marked by their initial points and directions, which areelements of the phase space, and there is a standard volume measure onthe phase space.On the sphere all geodesics (large circles) are periodic, but there are othermanifolds (f.e. Zoll-Tanner manifolds) with the same property.



The result of J. J. Duistermaat and Victor Guillemin is remarkable,because it connects the spectral properties of operator (quantum property)and the non-periodicity properties of the Hamiltonian (in this casegeodesic flow) Ψt (classical property).

But what about sharp remainder estimate for manifolds (or domains) withthe boundary? It is coming!



The result of J. J. Duistermaat and Victor Guillemin is remarkable,because it connects the spectral properties of operator (quantum property)and the non-periodicity properties of the Hamiltonian (in this casegeodesic flow) Ψt (classical property).But what about sharp remainder estimate for manifolds (or domains) withthe boundary? It is coming!



In 1978 Robert Seeley proved remainder estimateO(𝜆(d−1)/2) for compact domains (generalization tomanifolds with the boundary was easy).

He did not construct explicitly u(x , y , t) near the borderfor the wave equation in the general case (and nobody didso far!) but he invented some kind of the short dynamicsarguments which allowed him to get around this obstaclerather than overcome it.

Robert Seeley



In 1978 Robert Seeley proved remainder estimateO(𝜆(d−1)/2) for compact domains (generalization tomanifolds with the boundary was easy).He did not construct explicitly u(x , y , t) near the borderfor the wave equation in the general case (and nobody didso far!) but he invented some kind of the short dynamicsarguments which allowed him to get around this obstaclerather than overcome it.

Robert Seeley



In 1979 I proved the Weyl’s conjecture, recovering remainder estimateo(𝜆(d−1)/2) for compact manifolds with the boundary.

I also used the microlocal analysis, short dynamics arguments (but verydifferent from those of Seeley) and long dynamics arguments, similar tothose of Duistermaat and Guillemin. Sure, there was a new


The measure of the set all periodic geodesic billiard trajectories is zero.

Geodesic billiard trajectories consist of segments of geodesics, whichreflect from the boundary according to geometric optics law reflectionangle = incidence angle.However, billiard flow is much more complicated than the geodesic flow:billiards can behave badly, to be tangent to the boundary, or to makeinfinitely many jumps for a finite time. Points of the phase space, startingfrom each billiards behave badly (for positive or negative time) are calleddead-end points. Luckily, the set of all dead-end points has measure zero.



In 1979 I proved the Weyl’s conjecture, recovering remainder estimateo(𝜆(d−1)/2) for compact manifolds with the boundary.I also used the microlocal analysis, short dynamics arguments (but verydifferent from those of Seeley) and long dynamics arguments, similar tothose of Duistermaat and Guillemin.

Sure, there was a new






In 1979 I proved the Weyl’s conjecture, recovering remainder estimateo(𝜆(d−1)/2) for compact manifolds with the boundary.I also used the microlocal analysis, short dynamics arguments (but verydifferent from those of Seeley) and long dynamics arguments, similar tothose of Duistermaat and Guillemin. Sure, there was a new









Geodesic billiard trajectories consist of segments of geodesics, whichreflect from the boundary according to geometric optics law reflectionangle = incidence angle.

However, billiard flow is much more complicated than the geodesic flow:billiards can behave badly, to be tangent to the boundary, or to makeinfinitely many jumps for a finite time. Points of the phase space, startingfrom each billiards behave badly (for positive or negative time) are calleddead-end points. Luckily, the set of all dead-end points has measure zero.






Geodesic billiard trajectories consist of segments of geodesics, whichreflect from the boundary according to geometric optics law reflectionangle = incidence angle.However, billiard flow is much more complicated than the geodesic flow:billiards can behave badly, to be tangent to the boundary, or to makeinfinitely many jumps for a finite time. Points of the phase space, startingfrom each billiards behave badly (for positive or negative time) are calleddead-end points.

Luckily, the set of all dead-end points has measure zero.



In 1980 I conjectured that the non-periodicity condition holds for generalEuclidean domains. Now this conjecture is considered as one of the mostdifficult problems of the mathematical billiards theory.

It was proven for generic Euclidean domains and also for some specialclasses of domains.

If you are interested in this topic (and in the further developments–andthere were a lot!), you can find more in my talk 100 years of Weyl’s lawand in my article, also called 100 years of Weyl’s law.


http://weyl.math.toronto.edu/victor_ivrii_Publications/preprints/Talk_9.pdf

https://link.springer.com/article/10.1007/s13373-016-0089-y


In 1980 I conjectured that the non-periodicity condition holds for generalEuclidean domains. Now this conjecture is considered as one of the mostdifficult problems of the mathematical billiards theory.It was proven for generic Euclidean domains and also for some specialclasses of domains.

If you are interested in this topic (and in the further developments–andthere were a lot!), you can find more in my talk 100 years of Weyl’s lawand in my article, also called 100 years of Weyl’s law.


http://weyl.math.toronto.edu/victor_ivrii_Publications/preprints/Talk_9.pdf

https://link.springer.com/article/10.1007/s13373-016-0089-y

Equidistribution of eigenfunctions


In 1974 Alexander Shnirelman began to study the equidis-tribution of eigenfunctions (of the Laplacian on theclosed manifold). Recall that eigenfunctions are func-tions 𝜓n = 0 such that

−Δ𝜓n = 𝜆n𝜓n. (3) AlexanderShnirelman

Definition 2

Eigenfunctions are equidistributed if∫𝜔|𝜓n|2 dx → mes(𝜔)

mes(Ω)as n → ∞ (14)

for all n except those belonging to subsequence nk of the density 0.



Subsequence nk has density 0 if it becomes rarified on the long intervals:#{k : nk ≤ N} = o(N) as N → ∞.

In fact, Shnirelman was interested in the stronger property:equidistribution in the phase space (space of coordinates and momenta)rather than in the configuration space (space of coordinates only).

It turned out that the equidistribution (which is now called quantumergodicity) is due to the ergodicity of the classical dynamical system, thestudy of quantizations of classically chaotic systems is sometimes calledquantum chaos.



Subsequence nk has density 0 if it becomes rarified on the long intervals:#{k : nk ≤ N} = o(N) as N → ∞.In fact, Shnirelman was interested in the stronger property:equidistribution in the phase space (space of coordinates and momenta)rather than in the configuration space (space of coordinates only).

It turned out that the equidistribution (which is now called quantumergodicity) is due to the ergodicity of the classical dynamical system, thestudy of quantizations of classically chaotic systems is sometimes calledquantum chaos.



Recall that we have a geodesic flow Ψt . Due to Liouville’s theorem itpreserves the volume of the phase space: mes(Ψt(𝜔)) = mes(𝜔) for all t.

Definition 3

The flow Ψt is called ergodic if for almost all trajectories and all subsets 𝜔of the phase space Ω (with mes(𝜔) > 0) the time trajectory spends inside𝜔 (if we consider time interval [0,T ]), divided by T tends tomes(𝜔)/mes(Ω) as T → +∞.

Almost all = except of measure zero.

One can say that ergodicity means that almost every trajectory forgets thepast after a while.



Ergodicity has been a popular topic of the research and now theequidistribution also is. Since the above result has been generalized tomanifolds with the boundary:

Ergodicity and equidistribution

Ergodicity of the geodesic billiard flow =⇒ equidistribution of theeigenfunctions

and there is a little doubt that the converse result is also true, let usconsider ergodicity of the Euclidean billiards, and its relation tonon-periodicity.






and there is a little doubt that the converse result is also true,

let usconsider ergodicity of the Euclidean billiards, and its relation tonon-periodicity.






and there is a little doubt that the converse result is also true, let usconsider ergodicity of the Euclidean billiards, and its relation tonon-periodicity.



Obviously ergodicity =⇒ nonperiodicity but converse is not true.

Indeed, consider circular billiard or rectangular billiard:

𝛼

(a) Circular billiard

𝛼

(b) Rectangular billiard

Circular billiard trajectory is periodic if and only if 𝛼/2𝜋 = m/n is rational(irreducible); then billiard trajectory closes after n reflections and m turnsaround center; otherwise it fills densely (but not uniformly) the ringbetween two circles.



Obviously ergodicity =⇒ nonperiodicity but converse is not true.Indeed, consider circular billiard or rectangular billiard:

𝛼

(a) Circular billiard

𝛼

(b) Rectangular billiard

Circular billiard trajectory is periodic if and only if 𝛼/2𝜋 = m/n is rational(irreducible); then billiard trajectory closes after n reflections and m turnsaround center; otherwise it fills densely (but not uniformly) the ringbetween two circles.



Rectangular billiard trajectory (with sides a, b) is periodic if and only iftan(𝛼) : a/b = m/n is rational (irreducible); then billiard trajectory closesafter 2m reflections from the horizontal sides and 2m reflections from thevertical sides; otherwise it fills uniformly densely the whole rectangle.

In both case trajectory “remembers” 𝛼 and there is no ergodicity.



Figure: Bunimovich Stadium: billiard flow is ergodic:single trajectory is drawn

Still, some billiard trajectories (bouncing balls) are not dense and there is asequence of eigenfunctions which are not equidistributed. But the set ofsuch trajectories has measure 0 and the sequence of such eigenfunctionshas a density 0.


Can one hear the shape of the drum?


In 1966 Mark Kac asked: Can one (with the perfect hear-ing) hear the shape of the drum?

In other words, know-ing all the frequences (or, equivalently, all eigenvalues ofthe Dirichlet Laplacian) in domain X , can one restoreuniquely X (up to Euclidean movements).First, mathematicians tried to find spectral invariants–characteristics which must coincide for isospectral do-mains (domains with the same spectra).

Marc Kac

For this they used heat equation method: they considered

𝜎(t) := Tr(etΔ

)=

∑n

e−𝜆nt t > 0. (15)




In 1966 Mark Kac asked: Can one (with the perfect hear-ing) hear the shape of the drum? In other words, know-ing all the frequences (or, equivalently, all eigenvalues ofthe Dirichlet Laplacian) in domain X , can one restoreuniquely X (up to Euclidean movements).

First, mathematicians tried to find spectral invariants–characteristics which must coincide for isospectral do-mains (domains with the same spectra).

Marc Kac


𝜎(t) := Tr(etΔ

)=

∑n

e−𝜆nt t > 0. (15)




In 1966 Mark Kac asked: Can one (with the perfect hear-ing) hear the shape of the drum? In other words, know-ing all the frequences (or, equivalently, all eigenvalues ofthe Dirichlet Laplacian) in domain X , can one restoreuniquely X (up to Euclidean movements).First, mathematicians tried to find spectral invariants–characteristics which must coincide for isospectral do-mains (domains with the same spectra). Marc Kac


𝜎(t) := Tr(etΔ

)=

∑n

e−𝜆nt t > 0. (15)



It was known that

𝜎(t) = c0t−d/2 + c1t

(1−d)/2 + . . . as t → +0 (16)

Here (for d = 2) c0 is proportional to the area, c1 to the perimeter, c2Euler’s characteristic (connected to the number of holes), etc – so thoseare spectral invariants, and one with the perfect hearing can hear them.

But the final answer (given in 1992 by Gordon, Webb, and Wolpert) wasnegative:



It was known that

𝜎(t) = c0t−d/2 + c1t

(1−d)/2 + . . . as t → +0 (16)

Here (for d = 2) c0 is proportional to the area, c1 to the perimeter, c2Euler’s characteristic (connected to the number of holes), etc – so thoseare spectral invariants, and one with the perfect hearing can hear them.But the final answer (given in 1992 by Gordon, Webb, and Wolpert) wasnegative:



One cannot hear the shape of the drum

There exist isospectral but not isometric domains.

Figure: Example of two isospectral domains

Still plenty of questions remain: can we hear the shape of the convexdrum? . . .


Nodal lines

Nodal lines

In yearly 19-th century, a musician and physicist ErnstChlani made an experiment shown thus confirming anearlier observation of Robert Hooke. It is now calledChladni plates.

This experiment has been repeated many times (usuallyby physics professors trying to impress prospective stu-dents), with the violin bow replaced by an electric soundspeaker).

Ernst Chladni

Chladni plates


Nodal lines

When Chladni showed his experiment in Paris, Napoleon set a prize for thebest mathematical explanation.

Marie-Sophie Germain was the only person who sub-mitted the solution (with the correct approach) but itwas rejected: the judging commission felt that “the trueequations of the movement were not established,” eventhough “the experiments presented ingenious results.”After several attempts however she was able to establishthe corresponding equation and on 8 January 1816 shebecame the first woman to win a prize from the ParisAcademy of Sciences.

Marie-SophieGermain

Equation she presented led to the eigenvalue problem for the 4-th orderequation (

𝜕4x + 𝜕2x𝜕2y + 𝜕4y

)𝜓n = 𝜆𝜓n (17)

with two boundary conditons and the lines on the Chladni plates are nodallines on which eigenfunction 𝜓(x , y) vanishes.


Nodal lines


Marie-Sophie Germain was the only person who sub-mitted the solution (with the correct approach) but itwas rejected: the judging commission felt that “the trueequations of the movement were not established,” eventhough “the experiments presented ingenious results.”

After several attempts however she was able to establishthe corresponding equation and on 8 January 1816 shebecame the first woman to win a prize from the ParisAcademy of Sciences.

Marie-SophieGermain


𝜕4x + 𝜕2x𝜕2y + 𝜕4y

)𝜓n = 𝜆𝜓n (17)



Nodal lines



Marie-SophieGermain


𝜕4x + 𝜕2x𝜕2y + 𝜕4y

)𝜓n = 𝜆𝜓n (17)



Nodal lines



Marie-SophieGermain


𝜕4x + 𝜕2x𝜕2y + 𝜕4y

)𝜓n = 𝜆𝜓n (17)

with two boundary conditons

and the lines on the Chladni plates are nodallines on which eigenfunction 𝜓(x , y) vanishes.


Nodal lines



Marie-SophieGermain


𝜕4x + 𝜕2x𝜕2y + 𝜕4y

)𝜓n = 𝜆𝜓n (17)



Nodal lines

Later mathematical interest switched to a simpler problem for a vibratingmembrane: (

𝜕2x + 𝜕2y)𝜓n = −𝜆𝜓n in 𝒟, (18)

𝜓n|𝜕𝒟 = 0. (19)

The first question was how many nodal domains are there where a nodaldomain is a connected component of {(x , y) : 𝜓n(x , y) = 0}. Let thenumber of nodal domains be N(𝜓n).In one dimensional case 𝜓′′

n = −𝜆𝜓n, 𝜓n(0) = 𝜓n(a) = 0 the answer issimple: 𝜓n(x) = sin(𝜋nx/a) and N(𝜓n) = n.


Nodal lines


𝜕2x + 𝜕2y)𝜓n = −𝜆𝜓n in 𝒟, (18)

𝜓n|𝜕𝒟 = 0. (19)

The first question was how many nodal domains are there where a nodaldomain is a connected component of {(x , y) : 𝜓n(x , y) = 0}. Let thenumber of nodal domains be N(𝜓n).

In one dimensional case 𝜓′′n = −𝜆𝜓n, 𝜓n(0) = 𝜓n(a) = 0 the answer is

simple: 𝜓n(x) = sin(𝜋nx/a) and N(𝜓n) = n.


Nodal lines


𝜕2x + 𝜕2y)𝜓n = −𝜆𝜓n in 𝒟, (18)

𝜓n|𝜕𝒟 = 0. (19)

The first question was how many nodal domains are there where a nodaldomain is a connected component of {(x , y) : 𝜓n(x , y) = 0}. Let thenumber of nodal domains be N(𝜓n).In one dimensional case 𝜓′′

n = −𝜆𝜓n, 𝜓n(0) = 𝜓n(a) = 0 the answer issimple: 𝜓n(x) = sin(𝜋nx/a) and N(𝜓n) = n.


Nodal lines

But two-dimensional case is much more difficult! While we know thatN(𝜓1) = 1 and 2 ≤ N(𝜓n) ≤ n, it is the only easy result. Indeed, considerrectangular domain 𝒟 = {(x , y) : 0 ≤ x ≤ a, 0 ≤ y ≤ b}. Then there areeigenfunctions 𝜓(m,n)(x , y) = sin(𝜋mx/a) sin(𝜋ny/b) and the nodal linesand domains are shown here:

(we assume that the corresponding eigenvalue𝜆(m,n) = 𝜋2(m2/a2 + n2/b2) is simple).


Nodal lines

But if we slightly perturb the rectangular domain (so it will not be arectangular anymore) the number of the nodal domains decreases becausefor generic domains there are no intersections: each of them breaks in twopossible ways, opening a passage.

There are other problems: describe the size of the nodal set{(x , y) : 𝜓(x , y) = 0} (a spectacular progress here was recently made),how nodal lines meet the boundary and so on.


Exotic spectra Mixed spectra

Exotic spectra: mixed spectra

We considered the cases when the spectrum consists of the eigenvalues ofthe finite multiplicity (discrete spectrum). But in the self-adjoint operatorsof mathematical physics can have other kinds of the spectra.

Schrodinger operator H = − 12m~2Δ+ V

for the free particle (V = 0) has an absolutely continuous spectrum [0,∞).In the general case, it has an absolutely continuous spectrum [0,∞) andeigenvalues in (−∞, 0); if V decays fast at infinity, there are finite numberof eigenvalues, but for Coulomb potential these eigenvalues accumulate to−0.






for the free particle (V = 0) has an absolutely continuous spectrum [0,∞).

In the general case, it has an absolutely continuous spectrum [0,∞) andeigenvalues in (−∞, 0); if V decays fast at infinity, there are finite numberof eigenvalues, but for Coulomb potential these eigenvalues accumulate to−0.






for the free particle (V = 0) has an absolutely continuous spectrum [0,∞).In the general case, it has an absolutely continuous spectrum [0,∞) andeigenvalues in (−∞, 0); if V decays fast at infinity, there are finite numberof eigenvalues, but for Coulomb potential these eigenvalues accumulate to−0.



Dirac operator H =∑

𝜈 𝛾𝜈(−i~𝜕𝜈) + 𝛾0mc2 + V

for the free particle (V = 0) has an absolutely continuous spectrum(−∞,−mc2] ∩ [mc2,∞).

With the potential, decaying at infinity, it has a finite or infinite number ofeigenvalues in the spectral gap; they may accumulate only to the ends ofthe gap.



Dirac operator H =∑

𝜈 𝛾𝜈(−i~𝜕𝜈) + 𝛾0mc2 + V

for the free particle (V = 0) has an absolutely continuous spectrum(−∞,−mc2] ∩ [mc2,∞).With the potential, decaying at infinity, it has a finite or infinite number ofeigenvalues in the spectral gap; they may accumulate only to the ends ofthe gap.


Exotic spectra Band spectra

Band spectra

Schrodinger operator with periodic potential V

has a band spectrum: bands of absolutely continuous spectrum areseparated by spectral gaps.

In dimension d = 1 for generic V there is an infinite number of gaps.

Ford ≥ 2 there is only a finite number of gaps–Bethe-Sommerfeld conjecture,proved in full generality only about 10 years ago by Leonid Parnovski andAlexander Sobolev.

Adding another potential W decaying at infinity, one can place a finite orinfinite number of eigenvalues inside each spectral gap.



Band spectra

Schrodinger operator with periodic potential V

has a band spectrum: bands of absolutely continuous spectrum areseparated by spectral gaps.

In dimension d = 1 for generic V there is an infinite number of gaps. Ford ≥ 2 there is only a finite number of gaps–Bethe-Sommerfeld conjecture,proved in full generality only about 10 years ago by Leonid Parnovski andAlexander Sobolev.

Adding another potential W decaying at infinity, one can place a finite orinfinite number of eigenvalues inside each spectral gap.



Integrated Density of States

Instead of eigenvalue counting function N(𝜆) in this case use IntegratedDensity of States

N(𝜆) := limℓ→∞

N(𝜆, ℓX )

mes(ℓX )

where 0 ∈ X is an open bounded domain in Rd and N(𝜆, ℓX ) is aneigenvalue counting function for the same operator in ℓX (stretched X )with the Dirichlet boundary condition.



It turns out that there is a complete spectral asymptotics (even if V isonly almost periodic)

N(𝜆) ∼∞∑n=0

𝜅n𝜆d/2−n as 𝜆→ +∞.

Again, this is a rather new result, due to Leonid Parnovski and RomanShterenberg.

Leonid Roman Alexander

Parnovski Shterenberg Sobolev


Exotic spectra Landau levels

Landau levels

2D Schrodinger operator with constant magnetic fieldH = 1

2m (−i~𝜕1 − Bx2/2)2 + (−i~𝜕2 + Bx1/2)

2 + V

for a free particle (V = 0) has a pure point spectrum of infinitemultiplicity, consisting of Landau levels En := 1

2m (2n + 1)B~,n = 0, 1, 2, . . .:

Let V decay at infinity. Then the spectrum consists of eigenvalues en,k ,en,k → En as k → ∞:

depending on V these eigenvalues accumulate to Landau levels either fromthe left, or from the right, or from the left and right.Essential spectrum consists of Landau levels.



Landau levels


2m (−i~𝜕1 − Bx2/2)2 + (−i~𝜕2 + Bx1/2)

2 + V


2m (2n + 1)B~,n = 0, 1, 2, . . .:


depending on V these eigenvalues accumulate to Landau levels either fromthe left, or from the right, or from the left and right.

Essential spectrum consists of Landau levels.



Landau levels


2m (−i~𝜕1 − Bx2/2)2 + (−i~𝜕2 + Bx1/2)

2 + V


2m (2n + 1)B~,n = 0, 1, 2, . . .:


depending on V these eigenvalues accumulate to Landau levels either fromthe left, or from the right, or from the left and right.Essential spectrum consists of Landau levels.


Exotic spectra Ten martini problem

Classification of spectra

The classification of the spectrum is based on the spectral measure. Eachmeasure µ could be decomposed into the sum of three measures:

atomic measure, supported in the finite or enumerable number ofpoints (point spectrum) with 𝜇(𝜆) := 𝜇((−∞, 𝜆)) =

∑k,𝜆k<𝜆mk ,

absolute continuous measure with 𝜇(𝜆) =∫ 𝜆−∞ 𝜌(t) dt,

singular continuous measure with 𝜇(𝜆) being continuous, but with𝜇′(𝜆) = 0 almost everywhere.

And until recently in all “real life” examples the singular continuousspectrum was empty.



Ten martini problem

But there was a candidate, almost Mathieu operator which is the discreteSchrodinger operator (on Z)

(H𝜆,𝛼,𝜃𝜓)n = 𝜓n+1 + 𝜓n−1 + 2𝜆 cos(2𝜋(𝜃 + n𝛼))𝜓n, (20)

which appears in mathematical study of the quantum Hall effect.

For rational 𝛼 this operator would have a band spectrum, but for irrational𝛼 it was conjectured by Mark Kac that it has a spectrum which is Cantorset. And he offered ten bottles of Martini to one who would prove it!

In 2009 Arthur Avila and Svetlana Zhito-mirskaya solved this problem, proving that forall 𝜆 = 0 and irrational 𝛼 the spectrum is aCantor set. Arthur Svetlana

Avila Zhitomirskaya



Ten martini problem

But there was a candidate, almost Mathieu operator which is the discreteSchrodinger operator (on Z)

(H𝜆,𝛼,𝜃𝜓)n = 𝜓n+1 + 𝜓n−1 + 2𝜆 cos(2𝜋(𝜃 + n𝛼))𝜓n, (20)

which appears in mathematical study of the quantum Hall effect.For rational 𝛼 this operator would have a band spectrum, but for irrational𝛼 it was conjectured by Mark Kac that it has a spectrum which is Cantorset. And he offered ten bottles of Martini to one who would prove it!

In 2009 Arthur Avila and Svetlana Zhito-mirskaya solved this problem, proving that forall 𝜆 = 0 and irrational 𝛼 the spectrum is aCantor set. Arthur Svetlana

Avila Zhitomirskaya



Cantor set is a closed set which is nowhere dense. However such sets couldhave positive measures. It is known that the spectrum does not depend on𝜃.It is now known, that

For 0 < 𝜆 < 1, H𝜆,𝛼,𝜃 has surely purely absolutely continuousspectrum.

For 𝜆 = 1, H𝜆,𝛼,𝜃 has almost surely purely singular continuousspectrum. (It is not known whether eigenvalues can exist forexceptional parameters.)

For 𝜆 > 1, H𝜆,𝛼,𝜃 has almost surely pure point spectrum. It is knownthat almost surely can not be replaced by surely.



That’s all!

Thank you!!


etudes in spectral theory - victor...

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