the spectral theory of the neumann-poincare operator and plasmon resonance€¦ · contents •...

The spectral theory of the Neumann-Poincareoperator and plasmon resonance

Hyeonbae Kang(Inha University)

April, 2017, Emerging Topics in Optics, IMA

The spectral theory of the Neumann-Poincare operator and plasmon resonance Hyeonbae Kang (Inha University)

Contents

• Neumann-Poincare operator and plasmon resonance

• Why NP spectrum?

• Spectral properties of the electro-static NP operator

• Smooth domains• Domains with corners

• Spectral properties of elastic NP operator


Neumann-Poincare operator

K∂Ω[ϕ](x) =1

ωd

∫

∂Ω

〈y − x , νy〉

|x − y |dϕ(y) dσ(y)

The Neumann-Poincare (NP) operator is a boundary integral operatornaturally appearing when solving the Dirichlet (or Neumann) problem

using layer potentials.


• Single layer potential:

S∂Ω[ϕ](x) :=

∫

∂Ω

Γ(x − y)ϕ(y) dσ(y) , x ∈ Rd ,

where Γ(x) be the fundamental solution to the Laplacian, i.e.,

Γ(x) =

1

2πln |x | , d = 2 ,

−1

4π|x |, d = 3 ,

• Neumann boundary value problem:

∆u = 0 in Ω,∂u

∂ν= g on ∂Ω.


• Look for a solution u(x) = S∂Ω[ϕ](x).

• Jump relation of single layer potential:

∂

∂νS∂Ω[ϕ]

∣∣±(x) =

(±1

2I +K∗

∂Ω

)[ϕ](x), x ∈ ∂Ω ,

where the operator K∂Ω is defined by

K∂Ω[ϕ](x) =1

ωd

∫

∂Ω

〈y − x , νy〉

|x − y |dϕ(y) dσ(y) , x ∈ ∂Ω,

and K∗∂Ω is its L2-adjoint.

• K∗∂Ω (or K∂Ω) is called the Neumann-Poincare (NP) operator associated

with the domain Ω.

• It carries geometric information of ∂Ω.


• Neumann (1887/1888), Poincare’s variational problem (1896).

• If ∂Ω is C1,α for some α > 0, K∗∂Ω is a compact operator on L2(∂Ω) (and

on H−1/2(∂Ω)) and the Fredholm theory developed (1903).

• If ∂Ω is Lipschitz, then K∗∂Ω is a singular integral operator (20th Century.

Marcinkiewicz, Zygmund, Calderon, Coifman, Meyer, Kenig, ... ).

• L2-boundedness (Coifman-McIntosh-Meyer, ’82)

• Invertibility of − 12 I +K∗

∂Ω (Verchota, ’84)• High precision computation (Roklin, Greengard, Helsing)

• Spectral theory

• Carleman (1916)• 1950’s, Alfors, Schiffer, ...


Symmetrization of the NP operator

• Plemelj’s symmetrization principle (Khavinson-Putinar-Shapiro ’07)

S∂ΩK∗∂Ω = K∂ΩS∂Ω.

• New inner product on H∗ := H−1/20 (∂Ω):

〈ϕ,ψ〉H∗ := −〈ϕ,S∂Ω[ψ]〉 = −

∫

∂Ω

∫

∂Ω

Γ(x − y)ϕ(x)ψ(y)dσ(x)dσ(y).

(H∗-norm is equivalent to H−1/2-norm (K-Kim-Lee-Shin, preprint).

• K∗∂Ω is self-adjoint on H∗

〈ϕ,K∗∂Ω[ψ]〉H∗ = −〈ϕ,S∂ΩK

∗∂Ω[ψ]〉 = −〈ϕ,K∂ΩS∂Ω[ψ]〉 = 〈K∗

∂Ω[ϕ], ψ〉H∗


Good news

20th century’s fundamental mathematics can be utilized: Among them are

• Functional analysis

• Spectral theory self-adjoint operator and pseudo-differential operator

• Theory of singular integral operator


Spectral property of self-adjoint operator

• Spectral resolution:

K∗∂Ω =

∫ 1/2

−1/2

t dE(t).

• If ∂Ω is C1,α, then K∗∂Ω is compact on H∗. So, K∗

∂Ω has eigenvaluesaccumulating to 0, and

K∗∂Ω =

∞∑

j=1

λjϕj ⊗ ϕj .

• Spectrum: absolutely continuous, singularly continuous, pure pointspectrum

σ = σac ∪ σsc ∪ σpp.


Why spectrum?

• Plasmon resonance on meta-materials

• Cloaking by anomalous localized resonance

• Stress concentration and field enhancement


Plasmon resonance

• Dielectric constant distribution

ǫ =

ǫc + iδ in Ω,

ǫm in Rd \ Ω.

ǫc < 0 (negative dielectric constant), δ: dissipation.

• Dielectric equation in quasi-static limit:

∇ · ǫ∇u = a · ∇δz in R

d ,

u(x) = O(|x |1−d ) as |x | → ∞,

where a is a constant vector and δz is the Dirac mass at z ∈ Rd \ Ω.


Plasmon resonance on meta-materials

• Representation of solution

uδ(x) = a · ∇xΓ(x − z) + S∂Ω[ϕδ ](x), x ∈ Rd ,

where the potential ϕδ ∈ H∗0 (∂Ω) is the solution to

(λI −K∗∂Ω) [ϕδ ] = ∂νFz on ∂Ω.

with

λ :=ǫc + ǫm + iδ

2(ǫc − ǫm) + 2iδ.

• λ→ǫc + ǫm

2(ǫc − ǫm)as δ → 0.

• ǫc is negative if and only ifǫc + ǫm

2(ǫc − ǫm)∈ (−1/2, 1/2). The NP spectrum

lies in (−1/2, 1/2).


Plasmon resonance

• Plasmon resonance in the quasi-static limit occurs at the NP spectrum.

• Perturbation formula for resonance frequencies on nano particles:Ammari- Millien-Ruiz-Zhang ’15, Ammari-Ruiz-Yu-Zhang, ’16


Cloaking by anomalous localized resonance (CALR)

1

1

−1+iδ

source

cloakingregion

plasmonicstructure

• If |z | > r∗ :=√

r3e /ri , then E(δ) = δ‖∇uδ‖2L2(Ω) → 0 as δ → 0.

• If |z | < r∗, then E(δ) → ∞ as δ → 0, and the source is cloaked(Milton-Nicorovici ’06)

* If |z | > r∗, then uδ → the solution without the structure (the structure iscloaked).


Cloaking by anomalous localized resonance (CALR)

• CALR is a resonance occurring at the limit point of NP eigenvalues(Ammari-Ciraolo-K-Lee-Milton ’13)

* Variational approach: Kohn-Lu-Schweizer-Weinstein ’13


Stress or field concentration

−3 −2 −1 0 1 2 3−1

0

1

2

3

4

5

6

7

8

9

10

• The field concentrates in between two inclusions with extremeconductivity (0 or ∞)

• It is because more and more eigenvalues of the corresponding NP operatorare getting closer to 1/2 and −1/2. (Bonnetier-Triki ’13, Lim-Yu ’14)


NP Spectrum on smooth domains

If a boundary is smooth, then the NP operator is compact operator andhas eigenvalues of finite multiplicities converging to 0


• If Ω is a disk, then 0 is the only eigenvalue of K∗∂Ω. Moreover, disks are

the only domain on which the NP operator is of finite rank (Shapiro)

• Concentric disks (Ammari-Ciraolo-K-Lee-Milton ’13)

• Eigenvalues are

±1

2

(

ri

re

)n

, n = 1, 2, . . .

and eigenfunctions are

[

0e±inθ

]

,

[

e±inθ

0

]

• CALR occurs at 0 (accumulation point of eigenvalues).• By adding another circle, eigenvalues are perturbed to be

non-zero and converges to 0.


• If Ω is an ellipse of the long axis a and short axis b, eigenvalues are

±1

2(a − b

a + b)n, n = 1, 2, . . . ,

and the corresponding eigenfunctions are elliptic harmonics. (The NPeigenvalues converge to 0 exponentially fast.)


Theorem (Miyanishi-Suzuki, ’16)

If Ω is a bounded domain in R2 with C k boundary, then for any α > −k + 3/2

λn = o(nα) as n → ∞.

• If ∂Ω is smooth, then λn = o(n−k) for any positive integer k .

• What about domains with real analytic boundaries? 3D?


Theorem (Ando-K-Miyanishi, ’16)

Let Ω be a bounded planar domain with the analytic boundary ∂Ω and ǫ∂Ω be

the modified maximal Grauert radius of ∂Ω. Let λn∞

n=1 be the eigenvalues of

the NP operator K∗∂Ω on H

−1/20 (∂Ω) enumerated in descending order. For any

ǫ < ǫ∂Ω there is a constant C such that

|λ2n−1| = |λ2n| ≤ Ce−nǫ (1)

for any n.

• The decay rate is optimal.

• modified maximal Grauert radius of ∂Ω is determined by the maximal setto which the defining function is analytically extended.


Ball in 3D:

• The NP eigenvalues are

1

2(2n + 1), n = 1, 2, . . . ,

and the corresponding eigenfunctions are the spherical harmonics of ordern (the multiplicities are 2n + 1).

• CALR does not occur due to slow convergence of eigenvalues:Ammari-Ciraolo-K-Lee-Milton ’14, Ando-K ’15

• Miyanishi-Suzuki, ’16: On smooth domains in 3D,

λn = o(n−p), p <1

2.

• The optimal rate seems to be n−1/2


NP Spectrum on domains with corners

If a domain has a corner, then the NP operator is a singular integraloperator and is not compact.


• Bounds on the essential spectrum of the NP operator on planarcurvilinear polygonal domains are obtained (Perfekt-Putinar 14).

bess = maxj

(1−

θjπ

)

Essential spectrum (invariant under unitary transformations): continuous spectrum +limit points of eigenvalues.


Intersecting disks (K-Lim-Yu ’15)

r

r

G2 = 8Θ = -Θ0<

G1 = 8Θ = Θ0<

W

2 Θ0

* Carleman’s thesis (1916)* Lei et al, ACS Nano (2011)


• Complete spectral resolution of the NP operator is derived.

• Spectrum:

σac (K∗∂Ω) = [−

1

2+θ0π,1

2−θ0π], σsc(K

∗∂Ω) = ∅, σpp(K

∗∂Ω) = ∅.

• The bound 12− θ0

πis exactly the one found by PP.

• At the continuous spectrum, resonance is stronger (or equal to) thanδ−1/2 and weaker than δ−1

• If the derivative of the spectral measure is continuous and nonzero,resonance is at the rate of δ−1/2.

• This is the first example of domains with corners where completespectrum is known


Classification of spectrum by resonance

Helsing-K-Lim ’16

• Aim: to show existence (or non-existence) of continuous spectrum & purepoint spectrum on domains with corners.

• Idea: Different resonance occurs at continuous spectrum & pure pointspectrum.

• A special technique is required for high precision computation on domainswith corners.

• Resonance rate:

α(t) = − limδ→0

log ‖ϕt,δ‖2∗

log δ

where ϕt,δ is the solution to

((t + iδ)I −K∗

∂Ω

)[ϕt,δ] = f


Classification of spectrum by resonance

Theorem (Helsing-K-Lim)

(i) If α(t) > 0, then t ∈ σ(K∗∂Ω).

(ii) If1

2≤ α(t) < 1, then t ∈ σc(K

∗∂Ω).

(iii) If α(t) = 1 and t is isolated, then t ∈ σpp(K∗∂Ω).


• Rectangles:

• What is the threshold r∗ = 2.201592 of the aspect ratio?• What is the relation between aspect ratio and the number of

eigenvalues?

Theorem (Helsing-K-Lim)

As the aspect ratio tends to ∞, the spectral bound tends to 1/2. So eigenvalue

exists.

• Essential spectrum seems to be [−bess , bess ].


Continuous spectrum and discrete spectrum: Converge? Some weak sense?

−1 −0.5 0 0.5 1

−10

−8

−6

−4

−2

0

2

4

6

8

10

superellipse |r1|k+|r

2|k=1 with k=10

12

x

α(x

)

−1 −0.5 0 0.5 1

−10

−8

−6

−4

−2

0

2

4

6

8

10

The square: polarizability

x

α+(x

)

realα+(x)

imagα+(x)

Figure: Polarizablity tensor (J. Helsing)


Theorem (Perfekt-Putinar, ’16)

σess = [−bess , bess ].

* Bonnetier-Zhang (’17) found a new proof based on Weyl sequences ofgeneralized eigenfunctions.


Continuous spectrum in 3D (Helsing-Putinar ’17)

NP Spectrum on a surface of revolution:

continuous spectrum + eigenvalues


Elastic NP operators

The elastic NP operator is not compact even on smooth domains.


• Kelvin matrix of fundamental solutions to the Lame operator:Γ = (Γij)

d

i,j=1,

Γij(x) =

−α1

4π

δij|x|

−α2

4π

xixj

|x|3, if d = 3,

α1

2πδij ln |x| −

α2

2π

xixj

|x|2, if d = 2,

where

α1 =1

2

(1

µ+

1

2µ+ λ

)and α2 =

1

2

(1

µ−

1

2µ+ λ

).

• The elastic NP operator:

K[f](x) := p.v.

∫

∂Ω

∂νyΓ(x− y)f(y)dσ(y) a.e. x ∈ ∂Ω,

where∂νu := λ(∇ · u)n+ 2µ(∇u)n on ∂Ω.


Elastic NP operator

Polynomial compactness: Let

k∗ :=µ

2µ+ λ.

• In 2D, K2 − k2∗I is compact (Ando-Ji-K-Kim-Yu, ’15).

• In 3D, K3 − k2∗K is compact (Ando-K-Miyanishi, ’17)

Spectral structure:

• K on 2D smooth domains has eigenvalues converging to ±k0.

• K on 3D has eigenvalues converging to 0, ±k∗.

* Analysis of CALR type resonance: AJKKY ’15, Li-Liu ’16, Ando-K-Kim-Yu ’17.* Elastic NP eigenvalues on balls: Deng-Li-Liu ’17.


Thank you!


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