We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in
ℝ2d, 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.

1. Introduction

The Landau Hamiltonian describes a charged particle
moving in a plane, influenced by a constant magnetic field of strength B>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
Theorem 3.2 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 Section 2, we define the Landau Hamiltonian and
give some auxiliary results about its spectrum, eigenspaces, and Green
function.

We begin Section 3 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 in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>

We denote by x=(x1,…,x2d) a point in ℝ2d.
Let B>0 and denote by a→ the magnetic vector potentiala→(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 byL=(−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
operatorsLj=(−i(∂∂x2j−1,∂∂x2j)−(a2j−1,a2j))2,in the Hilbert spaces ℋj=L2(ℝ2).
Note that ℋ=⊗j=1dℋj,
andL=L1⊗I⊗(d−1)+I⊗L2⊗I⊗(d−2)+⋯+I⊗(d−1)⊗Ld.

2.1. Landau Levels

The spectrum of
each two-dimensional Landau Hamiltonian Lj 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
eigenvaluesΛκ^=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 of L 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∂∂zj=12(∂∂x2j−1−i∂∂x2j),∂∂z¯j=12(∂∂x2j−1+i∂∂x2j).

We define creation and annihilation operators 𝒬j*, 𝒬j as𝒬j*=−2ie−W∂∂zjeW,𝒬j=−2ieW∂∂z¯je−W,and note that[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 function u 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

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
formu=∑|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
decompositionsℒΛμ=⨁|κ^|=μℒΛκ^,𝒫Λμ=⨁|κ^|=μ𝒫Λκ^hold in ℋ.

2.3. The Resolvent

Let Rρ=(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: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 kernel Gρ(x,y) of Rρ can be written as Gρ(x,y)=∫0∞e−ρte−Lt(x,y)dm(t).Now, since the variables
separate pairwise, we havee−Lt(x,y)=∏j=1de−Ljt(x2j−1,x2j,y2j−1,y2j).The formula for e−Ljt is given in [13]. It
readse−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) becomesGρ(x,y)=(B4π)dexp(−iB2∑j=1d(x2j−1y2j−x2jy2j−1))I(|x−y|2),whereI(s)=∫0∞e−ρt1sinhd(Bt/2)exp(−B4coth(Bt2)s)dm(t).An expansion of I(s) shows thatI(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 in <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M98"><mml:mrow><mml:msup><mml:mi>ℝ</mml:mi><mml:mrow><mml:mn>2</mml:mn><mml:mi>d</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula>

Let K⊂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) byLΩ=(−i∇−a→)2,inΩ, with magnetic Neumann
boundary conditions∂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 every μ∈ℕ and each ε, 0<ε<dB,
the number of eigenvalues of LΩ in the interval (Λμ,Λμ+ε) is finite.

Denote by l1(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.

Let μ∈ℕ.

If d=1 then limj→∞(j!(Λμ−lj(μ)))1/j=(B/2)(Cap(K))2.

If d>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 operators L 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˜ asL˜=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 operators L and L˜ have different domains, we cannot compare them
directly. However, they act in the same Hilbert space, so we can compare their
inverses. LetR=R0=L−1,R˜=L˜−1=LK−1⊕LΩ−1,and setV=R˜−R,Tμ=𝒫ΛμV𝒫Λμ,forμ∈ℕ.Lemma 3.3.

V is nonnegative and compact.

Proof.

See Section 4.1.

By Weyl's theorem, the essential spectrum of R 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.

Let τ>0 be such that ((Λμ−1−2τ,Λμ−1+2τ)∖{Λμ−1})∩σess(R)=∅.
Denote the eigenvalues of Tμ byt1(μ)≥t2(μ)≥⋯,and the eigenvalues of R˜ in the interval (Λμ−1,Λμ−1+τ) byr1(μ)≥r2(μ)≥⋯.Lemma 3.4.

Given ε>0, there exists an integer l such that(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
of R˜ 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 asSμU=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.

Let K0⋐K⋐K1 be compact domains such that ∂Ki∩Γ=∅.
There exist a constant C>0 and a subspace 𝒮⊂ℋ of finite codimension such
that1C〈f,SμK0f〉≤〈f,Tμf〉≤C〈f,SμK1f〉for all f∈𝒮.

Proof.

See Section 4.2.

The asymptotic expansion of the spectrum of SμU is given in the following lemma.
Lemma 3.6.

Denote by s1(μ)≥s2(μ)≥⋯ the eigenvalues of SμU and by n(λ,SμU) the number of eigenvalues of SμU greater than λ (counting multiplicity). Then

if d=1 we have limj→∞(j!sj(μ))1/j=(B/2)(Cap(U))2,

if d>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 Theorem
3.2. 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 satisfylimj→∞(j!tj(μ))1/j=B2(Cap(K))2if d=1,
andn(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˜ satisfylimj→∞(j!(rj(μ)−Λμ−1))1/j=B2(Cap(K))2if d=1,
andN(Λμ−1+λ,Λμ−1−1,R˜)~(μ+d−1d−1)1d!(|lnλ|ln|lnλ|)d,asλ↘0.

If we translate this in terms of L˜ we getlimj→∞(j!(Λμ−lj(μ)))1/j=B2(Cap(K))2for d=1,
andN(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 Lemmas
3.3 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 Lemma <xref ref-type="statement" rid="lem3.3">3.3</xref>

The operators L 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 of V.
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〈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 function χ∈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 Lemma <xref ref-type="statement" rid="lem3.5">3.5</xref>

We start by showing that Tμ,
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 that1C∥f∥L2(Γ)∥f∥H1(Γ)≤〈f,Tμf〉≤C∥f∥L2(Γ)∥f∥H1(Γ)for all f in that subspace.

Let f and g belong to ℋ.
Also, let u=Rf, v=R˜g, and w=Rg.
We saw in (4.1) that〈f,Vg〉=∫Γ∂Nu⋅(vΩ−vK)¯dS.

To go further we will introduce the
Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. Let Gρ(x,y) be as in (2.16). We start with the single- and
double-layer integral operators, defined by𝒜α(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 Γ𝒜αK=AαK,ℬαK=12α+Bα,𝒜αΩ=AαΩ,ℬαΩ=−12α+Bα.Using a Green-type formula for L in K we see thatβ=ℬβK−𝒜(∂NβK).If we combine this with the
limit relations (4.5), we get(B−12I)βK=A(∂NβK),onΓ.

A similar calculation
for Ω gives(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 in K and Ω as(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 that A 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 (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 of V.
We have〈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〈f,Vg〉=(Λμ)−2∫Γ∂Nf⋅(((ND)K−(ND)Ω)(∂Ng))¯dSor with the introduced operators
above〈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 problemLμf=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.

Let Bμ be the double-layer operator evaluated at the
boundary,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 formulaf(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∥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
when K 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{(w1,…,wd)∣wj=tzj,t∈ℂ,|t|<1}is a subset of K∘.
If the setlog|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 byVK(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{∥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.

Let d>1 and ω=κ^/|κ^|.
Then(κ^!sκ^)1/|κ^|~B2exp(2VK(ω))(1+o(1)),as|κ^|→∞.

Proof.

The denominator in (5.4) is easily
calculated to be ∥zκ^∥ℱB22=(2πB)d(2B)|κ^|κ^!.For the numerator, we do estimations
from above and below, as in [12]. First, note thatIκ^=∥zκ^∥ℋ˜2=∫log|K|exp(2〈κ^,x〉)dm˜(x),where dm˜(x) is the transformed measure. It is clear
thatIκ^≤exp(2|κ^|VK(ω))m(K).For the inequality in the other
direction, fix δ>0.
The hyperplane〈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 haveIκ^≥∫Pδexp(2|κ^|(1−δ)VK(ω))dm˜(x)≥Cδexp(2|κ^|(1−δ)VK(ω)),where Cδ=∫Pδdm˜(x)>0.
It follows that(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.

Acknowledgment

The author would
like to thank his supervisor, Professor Grigori Rozenblum, for introducing him
to this problem and for giving him all the support he needed.

FockV.Bemerkung zur Quantelung des harmonischen Oszillators im MagnetfeldLandauL.Diamagnetismus der MetalleHornbergerK.SmilanskyU.Magnetic edge statesPushnitskiA.RozenblumG.Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domainRaĭkovG. D.Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tipsRaikovG. D.WarzelS.Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentialsRaikovG. D.WarzelS.Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentialsMelgaardM.RozenblumG.Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rankRaikovG. D.Spectral asymptotics for the perturbed 2D Pauli operator with oscillating
magnetic fields. I: non-zero mean value of the magnetic fieldFilonovN.PushnitskiA.Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domainsRozenblumG.SobolevA. V.Discrete spectrum distribution of the landau operator perturbed by an expanding electric potentialParfënovO. G.The singular values of the imbedding operators of some classes of analytic functions of several variablesSimonB.LandkofN. S.BirmanM. S.SolomjakM. Z.TaylorM. E.AgranovichM. S.KatsenelenbaumB. Z.SivovA. N.VoitovichN. N.TaylorM. E.