# Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian

Authors: Mikael Persson
Journal: Advances in Mathematical Physics
 (2009)

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## Abstract

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain inℝ2d, d≥1. The spectrum of this operator consists of clusters of eigenvalues around the Landau levels. We
give asymptotic formulas for the rate of accumulation of eigenvalues in these clusters. When the compact is a Reinhardt domain we are able to show a more precise asymptotic formula.


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## Body

## 1. Introduction

The Landau Hamiltonian describes a charged particle
moving in a plane, influenced by a constant magnetic field of strengthB>0 orthogonal to the plane. It is a classical
result, see
[1, 2], that the spectrum of the Landau Hamiltonian
consists of infinitely degenerate eigenvalues B(2q+1), q=0,1,2,…,
called Landau levels.In this paper, we will study the even-dimensional Landau
Hamiltonian outside a compact obstacle, imposing magnetic Neumann conditions at
the boundary. Our motivation to study this operator comes mainly from the
papers [3, 4]. Spectral properties of
the exterior Landau Hamiltonian in the plane are discussed in
[3], under both Dirichlet and
Neumann conditions at the boundary, with focus mainly on properties of the
eigenfunctions. A more qualitative study of the spectrum is done in [4], where the authors fix an
interval around a Landau level and describe how fast the eigenvalues in that
cluster converge to that Landau level. They work in the plane and with
Dirichlet boundary conditions only. The goal of this paper is to perform the
same qualitative description when we impose magnetic Neumann conditions at the
boundary. Moreover, we do not limit ourself to the plane but work in arbitrary
even-dimensional Euclidean space.The result is that the eigenvalues do accumulate with
the same rate to the Landau levels for both types of boundary conditions; see
Theorem3.2 for the details. However, the eigenvalues can only accumulate to a
Landau level from below in the Neumann setting. In the Dirichlet case they
accumulate only from above.It should be mentioned that we suppose that the
compact set removed has no holes and that its boundary is smooth. This is far
more restrictive than the conditions imposed on the compact set in [4].Several different perturbations of the Landau
Hamiltonian have been studied in the last years; see [4–11]. They all share the
common idea of making a reduction to a certain Toeplitz-type operator    whose spectral asymptotics are known. We also do this kind of reduction. The
method we use is based on the theory for pseudodifferential operators and
boundary PDE methods, which we have not seen in any of the mentioned papers.In Section2, we define the Landau Hamiltonian and
give some auxiliary results about its spectrum, eigenspaces, and Green
function.We begin Section3 by defining the exterior Landau
Hamiltonian with magnetic Neumann boundary condition and formulating and
proving the main theorems
(Theorems 3.1
and 3.2) about the spectral asymptotics
of the operator. The main part of the proof, the reduction step, is quite
technical and therefore moved to Section
4. When the reduction step is done we
use the asymptotic formulas of the spectrum of the Toeplitz-type operators,
given in [8, 10], to obtain the asymptotic formulas in
Theorem 3.2.In the higher dimensional case (ℝ2d, d>1), we also consider the case when the compact
obstacle is a Reinhardt domain. We use some ideas from [12] to prove a more precise
asymptotic formula for the eigenvalues. This is done in Section 5.


## 2. The Landau Hamiltonian inℝ2d

We denote byx=(x1,…,x2d) a point in ℝ2d.
Let B>0 and denote by a→ the magnetic vector potential(2.1)a→(x)=(a1(x),…,a2d(x))=B2(−x2,x1,−x4,x3,…,−x2d,x2d−1).It corresponds to an isotropic
magnetic field of constant strength B.
The Landau Hamiltonian L in ℝ2d describes a charged, spinless particle in this
homogeneous magnetic field. It is given by(2.2)L=(−i∇−a→)2and is essentially self-adjoint
on the set C0∞(ℝ2d) in the usual Hilbert space ℋ=L2(ℝ2d).
For j=1,…,d, we also introduce the self-adjoint
operators(2.3)Lj=(−i(∂∂x2j−1,∂∂x2j)−(a2j−1,a2j))2,in the Hilbert spaces ℋj=L2(ℝ2).
Note that ℋ=⊗j=1dℋj,
and(2.4)L=L1⊗I⊗(d−1)+I⊗L2⊗I⊗(d−2)+⋯+I⊗(d−1)⊗Ld.


### 2.1. Landau Levels

The spectrum of
each two-dimensional Landau HamiltonianLj consists of the so-called Landau levels,
eigenvalues B(2q+1), q∈ℕ:={0,1,2,…},
each of infinite multiplicity. Let κ^=(κ1,…κd)∈ℕd be a multi-index. We  denote by |κ^|=κ1+⋯+κd the length of the multi-index κ^ and also set κ^!=κ1!⋯κd!.
From (2.4) it follows that the spectrum of L consists of the infinitely degenerate
eigenvalues(2.5)Λκ^=B∑j=1d(2κj+1),κj∈ℕ.Note that Λκ^=Λκ^' if |κ^|=|κ^'|.
Hence the spectrum of L consists of eigenvalues of the form Λμ=B(2μ+d), μ∈ℕ.


### 2.2. Creation and Annihilation Operators

The structure
of the eigenspaces ofL has been described before in
[8]. We give the results without
proofs. It is convenient to introduce complex notation. Let z=(z1,…,zd)∈ℂd,
where zj=x2j−1+ix2j.
Also, we use the scalar potential W(z)=−(B/4)|z|2 and the complex derivatives(2.6)∂∂zj=12(∂∂x2j−1−i∂∂x2j),∂∂z¯j=12(∂∂x2j−1+i∂∂x2j).We define creation and annihilation operators𝒬j*, 𝒬j as(2.7)𝒬j*=−2ie−W∂∂zjeW,𝒬j=−2ieW∂∂z¯je−W,and note that(2.8)[𝒬j*,𝒬k*]=[𝒬j,𝒬k]=[𝒬j*,𝒬k]=0,ifj≠k. The notation 𝒬j* for the creation operators is motivated by the
fact that it is the formal adjoint of 𝒬j in ℋ.A functionu belongs to the lowest Landau level Λ0 if and only if 𝒬ju=0 for j=1,…,d.
This means that the function f=e−Wu is an entire function, so via multiplication
by e−W the eigenspace ℒΛ0 corresponding to Λ0 is equivalent to the Fock
space(2.9)ℱB2={f∣fisentireand∫ℂd|f|2e−(B/2)|z|2dm(z)<∞}.Here, and elsewhere, dm denotes the Lebesgue measure. A function u belongs to the eigenspace ℒΛμ of the Landau level Λμ if and only if it can be written in the
form(2.10)u=∑|κ^|=μcκ^(𝒬*)κ^(eWfκ^),where (𝒬*)κ^=(𝒬1*)κ1⋯(𝒬d*)κd and fκ^ all belong to ℱB2.
The multiplicity of each eigenvalue Λμ is infinite. We denote by 𝒫Λκ^ and 𝒫Λμ the projection onto the eigenspaces ℒΛκ^ and ℒΛμ,
respectively, and note by
(2.16) that the orthogonal
decompositions(2.11)ℒΛμ=⨁|κ^|=μℒΛκ^,𝒫Λμ=⨁|κ^|=μ𝒫Λκ^hold in ℋ.


### 2.3. The Resolvent

LetRρ=(L+ρI)−1 be the resolvent of L, ρ≥0.
An explicit formula of the kernel Gρ(x,y) of Rρ was given in
[3] for d=1.
In Section  4.2, we will use the behavior of Gρ(x,y) near the diagonal x=y,
given in the following lemma.Lemma 2.1.
Rρ is an integral operator with kernel Gρ(x,y) that has the following singularity at the
diagonal:(2.12)Gρ(x,y)~(1πlog(1|x−y|)+O(1),d=1,12π2|x−y|−2+O(log(1|x−y|)),d=2,Γ(d−1)2πd|x−y|2−2d+O(|x−y|4−2d),d>2,as |x−y|→0.Proof.
The kernelGρ(x,y) of Rρ can be written as (2.13)Gρ(x,y)=∫0∞e−ρte−Lt(x,y)dm(t).Now, since the variables
separate pairwise, we have(2.14)e−Lt(x,y)=∏j=1de−Ljt(x2j−1,x2j,y2j−1,y2j).The formula for e−Ljt is given in [13]. It
reads(2.15)e−Ljt=B4πexp(−iB2(x2j−1y2j−x2jy2j−1))1sinh(Bt/2)×exp(−B4coth(Bt2)((x2j−1−y2j−1)2+(x2j−y2j)2)).Hence
the formula for Gρ(x,y) becomes(2.16)Gρ(x,y)=(B4π)dexp(−iB2∑j=1d(x2j−1y2j−x2jy2j−1))I(|x−y|2),where(2.17)I(s)=∫0∞e−ρt1sinhd(Bt/2)exp(−B4coth(Bt2)s)dm(t).An expansion of I(s) shows that(2.18)I(s)~((2B)log(1s)+O(1),d=1,8B2s−1+O(log(1s)),d=2,(4B)dΓ(d−1)2s1−d+O(s2−d),d>2,ass→0, from which
(2.12)
follows.


## 2.1. Landau Levels

The spectrum of
each two-dimensional Landau HamiltonianLj consists of the so-called Landau levels,
eigenvalues B(2q+1), q∈ℕ:={0,1,2,…},
each of infinite multiplicity. Let κ^=(κ1,…κd)∈ℕd be a multi-index. We  denote by |κ^|=κ1+⋯+κd the length of the multi-index κ^ and also set κ^!=κ1!⋯κd!.
From (2.4) it follows that the spectrum of L consists of the infinitely degenerate
eigenvalues(2.5)Λκ^=B∑j=1d(2κj+1),κj∈ℕ.Note that Λκ^=Λκ^' if |κ^|=|κ^'|.
Hence the spectrum of L consists of eigenvalues of the form Λμ=B(2μ+d), μ∈ℕ.


## 2.2. Creation and Annihilation Operators

The structure
of the eigenspaces ofL has been described before in
[8]. We give the results without
proofs. It is convenient to introduce complex notation. Let z=(z1,…,zd)∈ℂd,
where zj=x2j−1+ix2j.
Also, we use the scalar potential W(z)=−(B/4)|z|2 and the complex derivatives(2.6)∂∂zj=12(∂∂x2j−1−i∂∂x2j),∂∂z¯j=12(∂∂x2j−1+i∂∂x2j).We define creation and annihilation operators𝒬j*, 𝒬j as(2.7)𝒬j*=−2ie−W∂∂zjeW,𝒬j=−2ieW∂∂z¯je−W,and note that(2.8)[𝒬j*,𝒬k*]=[𝒬j,𝒬k]=[𝒬j*,𝒬k]=0,ifj≠k. The notation 𝒬j* for the creation operators is motivated by the
fact that it is the formal adjoint of 𝒬j in ℋ.A functionu belongs to the lowest Landau level Λ0 if and only if 𝒬ju=0 for j=1,…,d.
This means that the function f=e−Wu is an entire function, so via multiplication
by e−W the eigenspace ℒΛ0 corresponding to Λ0 is equivalent to the Fock
space(2.9)ℱB2={f∣fisentireand∫ℂd|f|2e−(B/2)|z|2dm(z)<∞}.Here, and elsewhere, dm denotes the Lebesgue measure. A function u belongs to the eigenspace ℒΛμ of the Landau level Λμ if and only if it can be written in the
form(2.10)u=∑|κ^|=μcκ^(𝒬*)κ^(eWfκ^),where (𝒬*)κ^=(𝒬1*)κ1⋯(𝒬d*)κd and fκ^ all belong to ℱB2.
The multiplicity of each eigenvalue Λμ is infinite. We denote by 𝒫Λκ^ and 𝒫Λμ the projection onto the eigenspaces ℒΛκ^ and ℒΛμ,
respectively, and note by
(2.16) that the orthogonal
decompositions(2.11)ℒΛμ=⨁|κ^|=μℒΛκ^,𝒫Λμ=⨁|κ^|=μ𝒫Λκ^hold in ℋ.


## 2.3. The Resolvent

LetRρ=(L+ρI)−1 be the resolvent of L, ρ≥0.
An explicit formula of the kernel Gρ(x,y) of Rρ was given in
[3] for d=1.
In Section  4.2, we will use the behavior of Gρ(x,y) near the diagonal x=y,
given in the following lemma.Lemma 2.1.
Rρ is an integral operator with kernel Gρ(x,y) that has the following singularity at the
diagonal:(2.12)Gρ(x,y)~(1πlog(1|x−y|)+O(1),d=1,12π2|x−y|−2+O(log(1|x−y|)),d=2,Γ(d−1)2πd|x−y|2−2d+O(|x−y|4−2d),d>2,as |x−y|→0.Proof.
The kernelGρ(x,y) of Rρ can be written as (2.13)Gρ(x,y)=∫0∞e−ρte−Lt(x,y)dm(t).Now, since the variables
separate pairwise, we have(2.14)e−Lt(x,y)=∏j=1de−Ljt(x2j−1,x2j,y2j−1,y2j).The formula for e−Ljt is given in [13]. It
reads(2.15)e−Ljt=B4πexp(−iB2(x2j−1y2j−x2jy2j−1))1sinh(Bt/2)×exp(−B4coth(Bt2)((x2j−1−y2j−1)2+(x2j−y2j)2)).Hence
the formula for Gρ(x,y) becomes(2.16)Gρ(x,y)=(B4π)dexp(−iB2∑j=1d(x2j−1y2j−x2jy2j−1))I(|x−y|2),where(2.17)I(s)=∫0∞e−ρt1sinhd(Bt/2)exp(−B4coth(Bt2)s)dm(t).An expansion of I(s) shows that(2.18)I(s)~((2B)log(1s)+O(1),d=1,8B2s−1+O(log(1s)),d=2,(4B)dΓ(d−1)2s1−d+O(s2−d),d>2,ass→0, from which
(2.12)
follows.


## 3. The Exterior Landau-Neumann Hamiltonian inℝ2d

LetK⊂ℝ2d be a simply connected compact domain with
smooth boundary Γ and let Ω=ℝ2d∖K.
We define the exterior Landau-Neumann Hamiltonian LΩ in ℋΩ=L2(Ω) by(3.1)LΩ=(−i∇−a→)2,inΩ, with magnetic Neumann
boundary conditions(3.2)∂Nu:=(−i∇−a→)u⋅ν=0,onΓ.  Here ν denotes the exterior normal to Γ.
Our aim is to study how much the spectrum of LΩ differs from the Landau levels discussed in
the previous section. The first theorem below states that the eigenvalues of LΩ can accumulate to each Landau level only from
below. The second theorem says that the eigenvalues do accumulate to the Landau
levels from below, and the rate of convergence is given.Theorem 3.1.
For everyμ∈ℕ and each ε, 0<ε<dB,
the number of eigenvalues of LΩ in the interval (Λμ,Λμ+ε) is finite.Denote byl1(μ)≤l2(μ)≤⋯ the eigenvalues of LΩ in the interval (Λμ−1,Λμ) and by N(a,b,T) the number of eigenvalues of the operator T in the interval (a,b),
counting multiplicities. Also, let Cap(K) denote the logarithmic capacity of K;
see  [14, Chapter 2].Theorem 3.2.
Letμ∈ℕ.
(a)
Ifd=1 then  limj→∞(j!(Λμ−lj(μ)))1/j=(B/2)(Cap(K))2.(b)
Ifd>1 then N(Λμ−1,Λμ−λ,LΩ)~(μ+d−1d−1)(1/d!)(|lnλ|/ln|lnλ|)d as λ↘0.


### 3.1. Proof of The Theorems

We want to
compare the spectrum of the operatorsL and LΩ.
However, the expression L−LΩ has no meaning since L and LΩ act in
different Hilbert spaces. We introduce the Hilbert space ℋK=L2(K) and define the interior Landau-Neumann
Hamiltonian LK in ℋK by the same formulas as in
(3.1) and
(3.2) but
with Ω replaced by K.
We note that ℋ=ℋK⊕ℋΩ and define L˜ as(3.3)L˜=LK⊕LΩ,inℋK⊕ℋΩ.The inverse of LK is compact, so LK has at most a finite number of eigenvalues in
each interval (Λμ−1,Λμ).
The operators LK and LΩ act in orthogonal subspaces of ℋ,
so σ(L˜)=σ(LK)∪σ(LΩ).
This means that L˜ has the same spectral asymptotics as LΩ in each interval (Λμ−1,Λμ),
so it is enough to prove the statements in Theorems
3.1 and
3.2 for the
operator L˜ instead of LΩ.Since the unbounded operatorsL and L˜ have different domains, we cannot compare them
directly. However, they act in the same Hilbert space, so we can compare their
inverses. Let(3.4)R=R0=L−1,R˜=L˜−1=LK−1⊕LΩ−1,and set(3.5)V=R˜−R,Tμ=𝒫ΛμV𝒫Λμ,forμ∈ℕ.Lemma 3.3.
V is nonnegative and compact.Proof.
See Section4.1.By Weyl's theorem, the essential spectrum ofR and R˜ coincides. Since R˜=R+V and V≥0,
Theorem 3.1 follows immediately from [15, Theorem 9.4.7]  and the fact that σ(R)=σess(R)={Λμ−1}.
We continue with the proof of Theorem
3.2.Letτ>0 be such that ((Λμ−1−2τ,Λμ−1+2τ)∖{Λμ−1})∩σess(R)=∅.
Denote the eigenvalues of Tμ by(3.6)t1(μ)≥t2(μ)≥⋯,and the eigenvalues of R˜ in the interval (Λμ−1,Λμ−1+τ) by(3.7)r1(μ)≥r2(μ)≥⋯.Lemma 3.4.
Givenε>0, there exists an integer l such that(3.8)(1−ε)tj+l(μ)≤rj(μ)−Λμ−1≤(1+ε)tj−l(μ),for all sufficiently large j.Proof.
See [4, Proposition 2.2].Hence the study of the asymptotics of the eigenvalues
ofR˜ is reduced to the study of the eigenvalues of
the Toeplitz-type operator Tμ.
For a bounded simply connected set U in ℝ2d, we define the Toeplitz operator SμU as(3.9)SμU=𝒫ΛμχU𝒫Λμ,where χU denotes the characteristic function of U.
The following lemma reduces our problem to the study of these Toeplitz
operators, which are easier to study than Tμ.Lemma 3.5.
LetK0⋐K⋐K1 be compact domains such that ∂Ki∩Γ=∅.
There exist a constant C>0 and a subspace 𝒮⊂ℋ of finite codimension such
that(3.10)1C〈f,SμK0f〉≤〈f,Tμf〉≤C〈f,SμK1f〉for all f∈𝒮.Proof.
See Section4.2.The asymptotic expansion of the spectrum ofSμU is given in the following lemma.
Lemma 3.6.
Denote bys1(μ)≥s2(μ)≥⋯ the eigenvalues of SμU and by n(λ,SμU) the number of eigenvalues of SμU greater than λ (counting multiplicity). Then
(a)
ifd=1 we have limj→∞(j!sj(μ))1/j=(B/2)(Cap(U))2,(b)
ifd>1  we have n(λ,SμU)~(μ+d−1d−1)(1/d!)(|lnλ|/ln|lnλ|)d as λ↘0.Proof.
See [10, Lemma 3.2]
for part (a) and
[8, Proposition 7.1]
for part (b).Proof.
We are now able to finish the proof of Theorem3.2. By
letting K0 and K1 in Lemma 3.5 get closer and closer to our
compact K we see that the eigenvalues {tj(μ)} of Tμ satisfy(3.11)limj→∞(j!tj(μ))1/j=B2(Cap(K))2if d=1,
and(3.12)n(λ,Tμ)~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|)d,asλ↘0 if d>1.
Since neither formula
(3.11) nor
(3.12) is
sensitive for finite shifts in the indices,
it follows from
Lemma
3.4
that the eigenvalues of {rj(μ)}R˜ satisfy(3.13)limj→∞(j!(rj(μ)−Λμ−1))1/j=B2(Cap(K))2if d=1,
and(3.14)N(Λμ−1+λ,Λμ−1−1,R˜)~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|)d,asλ↘0.
If we translate this  in terms ofL˜ we get(3.15)limj→∞(j!(Λμ−lj(μ)))1/j=B2(Cap(K))2for d=1,
and(3.16)N(Λμ−1,Λμ−λ,L˜)~(μ+d−1d−1)1d!(|ln(λ/Λμ(Λμ−λ))|ln|ln(λ/Λμ(Λμ−λ))|)d~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|),asλ↘0,
for d>1.
This completes the proof of Theorem
3.2.


## 3.1. Proof of The Theorems

We want to
compare the spectrum of the operatorsL and LΩ.
However, the expression L−LΩ has no meaning since L and LΩ act in
different Hilbert spaces. We introduce the Hilbert space ℋK=L2(K) and define the interior Landau-Neumann
Hamiltonian LK in ℋK by the same formulas as in
(3.1) and
(3.2) but
with Ω replaced by K.
We note that ℋ=ℋK⊕ℋΩ and define L˜ as(3.3)L˜=LK⊕LΩ,inℋK⊕ℋΩ.The inverse of LK is compact, so LK has at most a finite number of eigenvalues in
each interval (Λμ−1,Λμ).
The operators LK and LΩ act in orthogonal subspaces of ℋ,
so σ(L˜)=σ(LK)∪σ(LΩ).
This means that L˜ has the same spectral asymptotics as LΩ in each interval (Λμ−1,Λμ),
so it is enough to prove the statements in Theorems
3.1 and
3.2 for the
operator L˜ instead of LΩ.Since the unbounded operatorsL and L˜ have different domains, we cannot compare them
directly. However, they act in the same Hilbert space, so we can compare their
inverses. Let(3.4)R=R0=L−1,R˜=L˜−1=LK−1⊕LΩ−1,and set(3.5)V=R˜−R,Tμ=𝒫ΛμV𝒫Λμ,forμ∈ℕ.Lemma 3.3.
V is nonnegative and compact.Proof.
See Section4.1.By Weyl's theorem, the essential spectrum ofR and R˜ coincides. Since R˜=R+V and V≥0,
Theorem 3.1 follows immediately from [15, Theorem 9.4.7]  and the fact that σ(R)=σess(R)={Λμ−1}.
We continue with the proof of Theorem
3.2.Letτ>0 be such that ((Λμ−1−2τ,Λμ−1+2τ)∖{Λμ−1})∩σess(R)=∅.
Denote the eigenvalues of Tμ by(3.6)t1(μ)≥t2(μ)≥⋯,and the eigenvalues of R˜ in the interval (Λμ−1,Λμ−1+τ) by(3.7)r1(μ)≥r2(μ)≥⋯.Lemma 3.4.
Givenε>0, there exists an integer l such that(3.8)(1−ε)tj+l(μ)≤rj(μ)−Λμ−1≤(1+ε)tj−l(μ),for all sufficiently large j.Proof.
See [4, Proposition 2.2].Hence the study of the asymptotics of the eigenvalues
ofR˜ is reduced to the study of the eigenvalues of
the Toeplitz-type operator Tμ.
For a bounded simply connected set U in ℝ2d, we define the Toeplitz operator SμU as(3.9)SμU=𝒫ΛμχU𝒫Λμ,where χU denotes the characteristic function of U.
The following lemma reduces our problem to the study of these Toeplitz
operators, which are easier to study than Tμ.Lemma 3.5.
LetK0⋐K⋐K1 be compact domains such that ∂Ki∩Γ=∅.
There exist a constant C>0 and a subspace 𝒮⊂ℋ of finite codimension such
that(3.10)1C〈f,SμK0f〉≤〈f,Tμf〉≤C〈f,SμK1f〉for all f∈𝒮.Proof.
See Section4.2.The asymptotic expansion of the spectrum ofSμU is given in the following lemma.
Lemma 3.6.
Denote bys1(μ)≥s2(μ)≥⋯ the eigenvalues of SμU and by n(λ,SμU) the number of eigenvalues of SμU greater than λ (counting multiplicity). Then
(a)
ifd=1 we have limj→∞(j!sj(μ))1/j=(B/2)(Cap(U))2,(b)
ifd>1  we have n(λ,SμU)~(μ+d−1d−1)(1/d!)(|lnλ|/ln|lnλ|)d as λ↘0.Proof.
See [10, Lemma 3.2]
for part (a) and
[8, Proposition 7.1]
for part (b).Proof.
We are now able to finish the proof of Theorem3.2. By
letting K0 and K1 in Lemma 3.5 get closer and closer to our
compact K we see that the eigenvalues {tj(μ)} of Tμ satisfy(3.11)limj→∞(j!tj(μ))1/j=B2(Cap(K))2if d=1,
and(3.12)n(λ,Tμ)~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|)d,asλ↘0 if d>1.
Since neither formula
(3.11) nor
(3.12) is
sensitive for finite shifts in the indices,
it follows from
Lemma
3.4
that the eigenvalues of {rj(μ)}R˜ satisfy(3.13)limj→∞(j!(rj(μ)−Λμ−1))1/j=B2(Cap(K))2if d=1,
and(3.14)N(Λμ−1+λ,Λμ−1−1,R˜)~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|)d,asλ↘0.
If we translate this  in terms ofL˜ we get(3.15)limj→∞(j!(Λμ−lj(μ)))1/j=B2(Cap(K))2for d=1,
and(3.16)N(Λμ−1,Λμ−λ,L˜)~(μ+d−1d−1)1d!(|ln(λ/Λμ(Λμ−λ))|ln|ln(λ/Λμ(Λμ−λ))|)d~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|),asλ↘0,
for d>1.
This completes the proof of Theorem
3.2.


## 4. Proof of The Lemmas

In this section, we prove Lemmas3.3 and
3.5. We will
use the theory of pseudodifferential operators and boundary layer potentials.
More details about these tools can be found in
[16]  and
[17, Chapter 5].


### 4.1. Proof of Lemma3.3

The operatorsL and L˜ are defined by the same expression, but the
domain of L˜ is contained in the domain of L.
It follows from
[4, Proposition 2.1]  that L−L˜≥0,
and hence V=R˜−R≥0.Next we prove the compactness ofV.
Let f and g belong to ℋ.
Also, let u=Rf and v=R˜g.
Then u belongs to the domain of L and v belongs to the domain of L˜,
so v=vK⊕vΩ,
and LKvK⊕LΩvΩ=g.
Integrating by parts and using
(3.2) for vK and vΩ,
we get(4.1)〈f,Vg〉=〈f,R˜g〉−〈Rf,g〉=∫KLu⋅vK¯dm(x)+∫ΩLu⋅vΩ¯dm(x)−∫Ku⋅LKvK¯dm(x)−∫Ωu⋅LΩvΩ¯dm(x)=∫Γ∂Nu⋅(vΩ−vK)¯dS.Here dS denotes the surface measure on Γ.Take a smooth cutoff functionχ∈C0∞(ℝ2d) such that χ(x)=1 in a neighborhood of K.
Then we can replace u and v by u˜=χu and v˜=χv in the right-hand side of
(4.1). By local
elliptic regularity we have that u˜∈H2(ℝ2d) and v˜∈H2(ℝ2d∖Γ).
However, the operator u˜↦∂Nu˜|Γ is compact as considered from H2(ℝ2d) to L2(Γ) and both v˜↦v˜Ω|Γ and v˜↦v˜K|Γ are compact as considered from H2(ℝ2d∖Γ) to L2(Γ),
so it follows that V is compact.


### 4.2. Proof of Lemma3.5

We start by showing thatTμ,
originally defined in L2(ℝ2d),
can be reduced to an operator in L2(Γ).
More precisely we show that Tμ can be realized as an elliptic
pseudodifferential operator of order 1 on some subspace of L2(Γ) of finite codimension, and hence there exists
a constant C>0 such that(4.2)1C∥f∥L2(Γ)∥f∥H1(Γ)≤〈f,Tμf〉≤C∥f∥L2(Γ)∥f∥H1(Γ)for all f in that subspace.Letf and g belong to ℋ.
Also, let u=Rf, v=R˜g, and w=Rg.
We saw in (4.1) that(4.3)〈f,Vg〉=∫Γ∂Nu⋅(vΩ−vK)¯dS.To go further we will introduce the
Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. LetGρ(x,y) be as in (2.16). We start with the single- and
double-layer integral operators, defined by(4.4)𝒜α(x)=∫ΓG0(x−y)α(y)dS(y),x∈ℝ2d,ℬα(x)=∫Γ∂NyG0(x−y)α(y)dS(y),x∈ℝ2d∖Γ,Aα(x)=∫ΓG0(x−y)α(y)dS(y),x∈Γ,Bα(x)=∫Γ∂NyG0(x−y)α(y)dS(y),x∈Γ.The last two operators are
compact on L2(Γ),
since, by Lemma 2.1, their kernels have weak singularities. Moreover, since the
kernel G0 has the same singularity as the Green kernel
for the Laplace operator in ℝ2d (see
[18, Chapter 7, Section 11]), we have the following
limit relations on Γ(4.5)𝒜αK=AαK,ℬαK=12α+Bα,𝒜αΩ=AαΩ,ℬαΩ=−12α+Bα.Using a Green-type formula for L in K we see that(4.6)β=ℬβK−𝒜(∂NβK).If we combine this with the
limit relations (4.5), we get(4.7)(B−12I)βK=A(∂NβK),onΓ.A similar calculation
forΩ gives(4.8)(B+12I)βΩ=A(∂NβΩ),onΓ.It seems natural to do
the following definitions.Definition 4.1.
We define the
Dirichlet-to-Neumann and Neumann-to-Dirichlet operators inK and Ω as(4.9)(DN)K=A−1(B−12I),(ND)K=(B−12I)−1A,(DN)Ω=A−1(B+12I),(ND)Ω=(B+12I)−1A.Remark 4.2.
The inverses above exist at least on a space of finite codimension. This
follows from the fact thatA is elliptic and B is compact.Lemma 4.3.
The operator(ND)K−(ND)Ω is an elliptic pseudodifferential operator of
order −1.Proof.
Using a resolvent identity, we see
that(4.10)(ND)K−(ND)Ω=(B+12I)−1(B−12I)−1A.It follows from the asymptotic
expansion of G0(x,y) in Lemma 2.1 and the fact that G0 is a Schwartz kernel (again, see
[18, Chapter 7,
Section 11])
that A is an elliptic pseudodifferential operator of
order −1.
Moreover the operator B is compact, so the other two factors are
pseudodifferential operators of order 0 which do not change the principal symbol
noticeably.Let us now return to the expression ofV.
We have(4.11)〈f,Vg〉=∫Γ∂Nu⋅(vΩ−vK)¯dS=∫Γ∂Nu⋅(vΩ−w+w−vK)¯dS=∫Γ∂Nu⋅((ND)Ω(∂N(vΩ−w)+(ND)K(∂N(w−vK))))¯dS=∫Γ∂Nu⋅(((ND)K−(ND)Ω)(∂Nw))¯dS.Since we are interested in Tμ and not V,
we may assume that f and g belong to ℒΛμ.
Then u=Rf=Λμ−1f and w=Rg=Λμ−1g.
For such f and g we get(4.12)〈f,Vg〉=(Λμ)−2∫Γ∂Nf⋅(((ND)K−(ND)Ω)(∂Ng))¯dSor with the introduced operators
above(4.13)〈f,Vg〉=(Λμ)−2∫Γf⋅((DN)K*((ND)K−(ND)Ω)((DN)Kg))¯dS.Moreover, (DN)K is an elliptic pseudodifferential operator of
order 1.
This follows from the identity A(DN)K=B−(1/2)I, and the fact that A is an elliptic pseudodifferential operator of
order −1.
It follows from (4.13) that Tμ is an elliptic pseudodifferential operator or
order 1.Next, we prove the inequality
(3.10). Because of the
projections, it is enough to show it for functions f in ℒΛμ.The lower bound. We prove that there exists a subspace 𝒮˜⊂ℒΛμ of finite codimension such that the lower
bound in (3.10) is valid for all f∈𝒮˜.
Since f∈ℒΛμ we have Lμf:=(L−Λμ)f=0 so f belongs to the kernel of the second-order
elliptic operator Lμ.
Let φ=f|Γ.
We study the problem(4.14)Lμf=0inK∘,f=φonΓ.Let E(x,y) be the Schwartz kernel for Lμ.
It is smooth away from the diagonal x=y.
One can repeat the theory with the single- and double-layer potentials for Lμ and write the solution f in the case it exists.LetBμ be the double-layer operator evaluated at the
boundary,(4.15)Bμα(x)=∫Γ∂NyE(x,y)α(y)dS(y),x∈Γ.The operator Bμ is compact, since the kernel ∂NyE(x,y) has a weak singularity at the diagonal x=y.
Thus there exists a subspace 𝒮1⊂L2(Γ) of finite codimension such that the operator (1/2)I+Bμ is invertible on 𝒮1.
Hence, there exists a subspace 𝒮˜⊂ℒΛμ of finite codimension where we have the
representation formula(4.16)f(x)=∫Γ∂E(x,y)∂νy((12I+Bμ)−1φ)(y)dS(y),x∈K∘for all f∈𝒮˜.
The inequality ∥f∥L2(K0)≤C∥f∥L2(Γ) follows easily from
(4.16) for all such
functions f.Since we also have∥f∥L2(Γ)≤C∥f∥H1(Γ) the lower bound in
(3.10) follows via the lower
bound in (4.2).The upper bound. By the upper bound in (4.2) it is enough to show the
following inequalities(4.17)∥f∥L2(Γ)∥f∥H1(Γ)≤C∥f∥H1/2(K)∥f∥H3/2(K)≤C∥f∥H2(K)2≤C∥f∥L2(K1)2.However, the first inequality is
just the Trace theorem, the second is the Sobolev-Rellich embedding theorem. We
note that Lμf=0,
so the third inequality is a standard estimate for elliptic operators.


## 4.1. Proof of Lemma3.3

The operatorsL and L˜ are defined by the same expression, but the
domain of L˜ is contained in the domain of L.
It follows from
[4, Proposition 2.1]  that L−L˜≥0,
and hence V=R˜−R≥0.Next we prove the compactness ofV.
Let f and g belong to ℋ.
Also, let u=Rf and v=R˜g.
Then u belongs to the domain of L and v belongs to the domain of L˜,
so v=vK⊕vΩ,
and LKvK⊕LΩvΩ=g.
Integrating by parts and using
(3.2) for vK and vΩ,
we get(4.1)〈f,Vg〉=〈f,R˜g〉−〈Rf,g〉=∫KLu⋅vK¯dm(x)+∫ΩLu⋅vΩ¯dm(x)−∫Ku⋅LKvK¯dm(x)−∫Ωu⋅LΩvΩ¯dm(x)=∫Γ∂Nu⋅(vΩ−vK)¯dS.Here dS denotes the surface measure on Γ.Take a smooth cutoff functionχ∈C0∞(ℝ2d) such that χ(x)=1 in a neighborhood of K.
Then we can replace u and v by u˜=χu and v˜=χv in the right-hand side of
(4.1). By local
elliptic regularity we have that u˜∈H2(ℝ2d) and v˜∈H2(ℝ2d∖Γ).
However, the operator u˜↦∂Nu˜|Γ is compact as considered from H2(ℝ2d) to L2(Γ) and both v˜↦v˜Ω|Γ and v˜↦v˜K|Γ are compact as considered from H2(ℝ2d∖Γ) to L2(Γ),
so it follows that V is compact.


## 4.2. Proof of Lemma3.5

We start by showing thatTμ,
originally defined in L2(ℝ2d),
can be reduced to an operator in L2(Γ).
More precisely we show that Tμ can be realized as an elliptic
pseudodifferential operator of order 1 on some subspace of L2(Γ) of finite codimension, and hence there exists
a constant C>0 such that(4.2)1C∥f∥L2(Γ)∥f∥H1(Γ)≤〈f,Tμf〉≤C∥f∥L2(Γ)∥f∥H1(Γ)for all f in that subspace.Letf and g belong to ℋ.
Also, let u=Rf, v=R˜g, and w=Rg.
We saw in (4.1) that(4.3)〈f,Vg〉=∫Γ∂Nu⋅(vΩ−vK)¯dS.To go further we will introduce the
Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. LetGρ(x,y) be as in (2.16). We start with the single- and
double-layer integral operators, defined by(4.4)𝒜α(x)=∫ΓG0(x−y)α(y)dS(y),x∈ℝ2d,ℬα(x)=∫Γ∂NyG0(x−y)α(y)dS(y),x∈ℝ2d∖Γ,Aα(x)=∫ΓG0(x−y)α(y)dS(y),x∈Γ,Bα(x)=∫Γ∂NyG0(x−y)α(y)dS(y),x∈Γ.The last two operators are
compact on L2(Γ),
since, by Lemma 2.1, their kernels have weak singularities. Moreover, since the
kernel G0 has the same singularity as the Green kernel
for the Laplace operator in ℝ2d (see
[18, Chapter 7, Section 11]), we have the following
limit relations on Γ(4.5)𝒜αK=AαK,ℬαK=12α+Bα,𝒜αΩ=AαΩ,ℬαΩ=−12α+Bα.Using a Green-type formula for L in K we see that(4.6)β=ℬβK−𝒜(∂NβK).If we combine this with the
limit relations (4.5), we get(4.7)(B−12I)βK=A(∂NβK),onΓ.A similar calculation
forΩ gives(4.8)(B+12I)βΩ=A(∂NβΩ),onΓ.It seems natural to do
the following definitions.Definition 4.1.
We define the
Dirichlet-to-Neumann and Neumann-to-Dirichlet operators inK and Ω as(4.9)(DN)K=A−1(B−12I),(ND)K=(B−12I)−1A,(DN)Ω=A−1(B+12I),(ND)Ω=(B+12I)−1A.Remark 4.2.
The inverses above exist at least on a space of finite codimension. This
follows from the fact thatA is elliptic and B is compact.Lemma 4.3.
The operator(ND)K−(ND)Ω is an elliptic pseudodifferential operator of
order −1.Proof.
Using a resolvent identity, we see
that(4.10)(ND)K−(ND)Ω=(B+12I)−1(B−12I)−1A.It follows from the asymptotic
expansion of G0(x,y) in Lemma 2.1 and the fact that G0 is a Schwartz kernel (again, see
[18, Chapter 7,
Section 11])
that A is an elliptic pseudodifferential operator of
order −1.
Moreover the operator B is compact, so the other two factors are
pseudodifferential operators of order 0 which do not change the principal symbol
noticeably.Let us now return to the expression ofV.
We have(4.11)〈f,Vg〉=∫Γ∂Nu⋅(vΩ−vK)¯dS=∫Γ∂Nu⋅(vΩ−w+w−vK)¯dS=∫Γ∂Nu⋅((ND)Ω(∂N(vΩ−w)+(ND)K(∂N(w−vK))))¯dS=∫Γ∂Nu⋅(((ND)K−(ND)Ω)(∂Nw))¯dS.Since we are interested in Tμ and not V,
we may assume that f and g belong to ℒΛμ.
Then u=Rf=Λμ−1f and w=Rg=Λμ−1g.
For such f and g we get(4.12)〈f,Vg〉=(Λμ)−2∫Γ∂Nf⋅(((ND)K−(ND)Ω)(∂Ng))¯dSor with the introduced operators
above(4.13)〈f,Vg〉=(Λμ)−2∫Γf⋅((DN)K*((ND)K−(ND)Ω)((DN)Kg))¯dS.Moreover, (DN)K is an elliptic pseudodifferential operator of
order 1.
This follows from the identity A(DN)K=B−(1/2)I, and the fact that A is an elliptic pseudodifferential operator of
order −1.
It follows from (4.13) that Tμ is an elliptic pseudodifferential operator or
order 1.Next, we prove the inequality
(3.10). Because of the
projections, it is enough to show it for functions f in ℒΛμ.The lower bound. We prove that there exists a subspace 𝒮˜⊂ℒΛμ of finite codimension such that the lower
bound in (3.10) is valid for all f∈𝒮˜.
Since f∈ℒΛμ we have Lμf:=(L−Λμ)f=0 so f belongs to the kernel of the second-order
elliptic operator Lμ.
Let φ=f|Γ.
We study the problem(4.14)Lμf=0inK∘,f=φonΓ.Let E(x,y) be the Schwartz kernel for Lμ.
It is smooth away from the diagonal x=y.
One can repeat the theory with the single- and double-layer potentials for Lμ and write the solution f in the case it exists.LetBμ be the double-layer operator evaluated at the
boundary,(4.15)Bμα(x)=∫Γ∂NyE(x,y)α(y)dS(y),x∈Γ.The operator Bμ is compact, since the kernel ∂NyE(x,y) has a weak singularity at the diagonal x=y.
Thus there exists a subspace 𝒮1⊂L2(Γ) of finite codimension such that the operator (1/2)I+Bμ is invertible on 𝒮1.
Hence, there exists a subspace 𝒮˜⊂ℒΛμ of finite codimension where we have the
representation formula(4.16)f(x)=∫Γ∂E(x,y)∂νy((12I+Bμ)−1φ)(y)dS(y),x∈K∘for all f∈𝒮˜.
The inequality ∥f∥L2(K0)≤C∥f∥L2(Γ) follows easily from
(4.16) for all such
functions f.Since we also have∥f∥L2(Γ)≤C∥f∥H1(Γ) the lower bound in
(3.10) follows via the lower
bound in (4.2).The upper bound. By the upper bound in (4.2) it is enough to show the
following inequalities(4.17)∥f∥L2(Γ)∥f∥H1(Γ)≤C∥f∥H1/2(K)∥f∥H3/2(K)≤C∥f∥H2(K)2≤C∥f∥L2(K1)2.However, the first inequality is
just the Trace theorem, the second is the Sobolev-Rellich embedding theorem. We
note that Lμf=0,
so the third inequality is a standard estimate for elliptic operators.


## 5. Spectrum of Toeplitz Operators in A Reinhardt Domain

In the case
whenK is a Reinhardt domain one can strengthen part (b) of Lemma 3.6. Assume that K∘,
the interior of K,
is a Reinhardt domain. This means that 0∈K∘ and if z∈K∘,
then the set(5.1){(w1,…,wd)∣wj=tzj,t∈ℂ,|t|<1}is a subset of K∘.
If the set(5.2)log|K|={(y1,…,yd)∣yj=log|zj|,z∈K∘}is convex in the usual sense,
then K∘ is said to be logarithmically convex, and K∘ is a domain of holomorphy. Denote by VK:ℝd→ℝ the function defined by(5.3)VK(x)=supy∈log|K|〈x,y〉.We denote by J:ℱB2→ℋ˜:=L2(K,e−(B/2)|z|2dm(z)) the embedding operator. The s -values sκ^, κ^∈ℕd,
of J
coincide
with the
numbers(5.4){∥zκ^∥ℋ˜2∥zκ^∥ℱB22}κ^≥0(we remind the reader of the
notation of eigenvalues in Lemma 3.6). Unlike the case d=1,
see [10], it is
natural to numerate the eigenvalues by the d-tuples κ^=(κ1,…,κd),
just as for the eigenvalues of the Laplace operator in the unit cube [0,1]d,
where the eigenvalues are given by (2π)−d|κ^|22=(2π)−d(κ12+⋯+κd2).Lemma 5.1.
Letd>1 and ω=κ^/|κ^|.
Then(5.5)(κ^!sκ^)1/|κ^|~B2exp(2VK(ω))(1+o(1)),as|κ^|→∞.Proof.
The denominator in (5.4) is easily
calculated to be (5.6)∥zκ^∥ℱB22=(2πB)d(2B)|κ^|κ^!.For the numerator, we do estimations
from above and below, as in [12]. First, note that(5.7)Iκ^=∥zκ^∥ℋ˜2=∫log|K|exp(2〈κ^,x〉)dm˜(x),where dm˜(x) is the transformed measure. It is clear
that(5.8)Iκ^≤exp(2|κ^|VK(ω))m(K).For the inequality in the other
direction, fix δ>0.
The hyperplane(5.9)〈κ^,x〉=(1−δ)VK(κ^)cuts log|K| in two components. Let Pδ be the component for which the inequality 〈κ^,x〉≥(1−δ)VK(κ^) holds. Then we have(5.10)Iκ^≥∫Pδexp(2|κ^|(1−δ)VK(ω))dm˜(x)≥Cδexp(2|κ^|(1−δ)VK(ω)),where Cδ=∫Pδdm˜(x)>0.
It follows that(5.11)(κ^!sκ^)1/|κ^|≤(m(K)(B2π)d)1/|κ^|B2exp(2VK(ω)),(κ^!sκ^)1/|κ^|≥(Cδ(B2π)d)1/|κ^|B2exp(2(1−δ)VK(ω)),from which
(5.5) follows.



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*Source: 873704-2008-11-25.xml*