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Advances in Mathematical Physics
Volumeย 2009, Article IDย 873704, 15 pages
http://dx.doi.org/10.1155/2009/873704
Research Article

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

Department of Mathematical Sciences, Chalmers University of Technology and Gรถteborg University, 412 96 Gรถteborg, Sweden

Received 27 June 2008; Accepted 22 October 2008

Academic Editor: Pavelย Exner

Copyright ยฉ 2009 Mikael Persson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study the Schrรถdinger operator with a constant magnetic field in the exterior of a compact domain in โ„2๐‘‘, ๐‘‘โ‰ฅ1. 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 ๐ต>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 ๐ต(2๐‘ž+1), ๐‘ž=0,1,2,โ€ฆ, 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 [4โ€“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 (โ„2๐‘‘, ๐‘‘>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 โ„2๐‘‘

We denote by ๐‘ฅ=(๐‘ฅ1,โ€ฆ,๐‘ฅ2๐‘‘) a point in โ„2๐‘‘. Let ๐ต>0 and denote by โ†’๐‘Ž the magnetic vector potentialIt corresponds to an isotropic magnetic field of constant strength ๐ต. The Landau Hamiltonian ๐ฟ in โ„2๐‘‘ describes a charged, spinless particle in this homogeneous magnetic field. It is given byand is essentially self-adjoint on the set ๐ถโˆž0(โ„2๐‘‘) in the usual Hilbert space โ„‹=๐ฟ2(โ„2๐‘‘). For ๐‘—=1,โ€ฆ,๐‘‘, we also introduce the self-adjoint operatorsin the Hilbert spaces โ„‹๐‘—=๐ฟ2(โ„2). Note that โ„‹=โŠ—๐‘‘๐‘—=1โ„‹๐‘—, and

2.1. Landau Levels

The spectrum of each two-dimensional Landau Hamiltonian ๐ฟ๐‘— consists of the so-called Landau levels, eigenvalues ๐ต(2๐‘ž+1), ๐‘žโˆˆโ„•โˆถ={0,1,2,โ€ฆ}, each of infinite multiplicity. Let ๎๐œ…=(๐œ…1,โ€ฆ๐œ…๐‘‘)โˆˆโ„•๐‘‘ be a multi-index. We denote by |๎๐œ…|=๐œ…1+โ‹ฏ+๐œ…๐‘‘ the length of the multi-index ๎๐œ… and also set ๎๐œ…!=๐œ…1!โ‹ฏ๐œ…๐‘‘!. From (2.4) it follows that the spectrum of ๐ฟ consists of the infinitely degenerate eigenvaluesNote that ฮ›๎๐œ…=ฮ›๎๐œ…โ€ฒ if |๎๐œ…|=|๎๐œ…๎…ž|. Hence the spectrum of ๐ฟ consists of eigenvalues of the form ฮ›๐œ‡=๐ต(2๐œ‡+๐‘‘), ๐œ‡โˆˆโ„•.

2.2. Creation and Annihilation Operators

The structure of the eigenspaces of ๐ฟ has been described before in [8]. We give the results without proofs. It is convenient to introduce complex notation. Let ๐‘ง=(๐‘ง1,โ€ฆ,๐‘ง๐‘‘)โˆˆโ„‚๐‘‘, where ๐‘ง๐‘—=๐‘ฅ2๐‘—โˆ’1+๐‘–๐‘ฅ2๐‘—. Also, we use the scalar potential ๐‘Š(๐‘ง)=โˆ’(๐ต/4)|๐‘ง|2 and the complex derivatives

We define creation and annihilation operators ๐’ฌโˆ—๐‘—, ๐’ฌ๐‘— asand note that The notation ๐’ฌโˆ—๐‘— for the creation operators is motivated by the fact that it is the formal adjoint of ๐’ฌ๐‘— in โ„‹.

A function ๐‘ข belongs to the lowest Landau level ฮ›0 if and only if ๐’ฌ๐‘—๐‘ข=0 for ๐‘—=1,โ€ฆ,๐‘‘. This means that the function ๐‘“=๐‘’โˆ’๐‘Š๐‘ข is an entire function, so via multiplication by ๐‘’โˆ’๐‘Š the eigenspace โ„’ฮ›0 corresponding to ฮ›0 is equivalent to the Fock space

Here, and elsewhere, d๐‘š denotes the Lebesgue measure. A function ๐‘ข belongs to the eigenspace โ„’ฮ›๐œ‡ of the Landau level ฮ›๐œ‡ if and only if it can be written in the formwhere (๐’ฌโˆ—)๎๐œ…=(๐’ฌโˆ—1)๐œ…1โ‹ฏ(๐’ฌโˆ—๐‘‘)๐œ…๐‘‘ and ๐‘“๎๐œ… all belong to โ„ฑ2๐ต. The multiplicity of each eigenvalue ฮ›๐œ‡ is infinite. We denote by ๐’ซฮ›๎๐œ… and ๐’ซฮ›๐œ‡ the projection onto the eigenspaces โ„’ฮ›๎๐œ… and โ„’ฮ›๐œ‡, respectively, and note by (2.16) that the orthogonal decompositionshold in โ„‹.

2.3. The Resolvent

Let ๐‘…๐œŒ=(๐ฟ+๐œŒ๐ผ)โˆ’1 be the resolvent of ๐ฟ, ๐œŒโ‰ฅ0. An explicit formula of the kernel ๐บ๐œŒ(๐‘ฅ,๐‘ฆ) of ๐‘…๐œŒ was given in [3] for ๐‘‘=1. In Section 4.2, we will use the behavior of ๐บ๐œŒ(๐‘ฅ,๐‘ฆ) near the diagonal ๐‘ฅ=๐‘ฆ, given in the following lemma.Lemma 2.1. ๐‘…๐œŒ is an integral operator with kernel ๐บ๐œŒ(๐‘ฅ,๐‘ฆ) that has the following singularity at the diagonal: as |๐‘ฅโˆ’๐‘ฆ|โ†’0.

Proof. The kernel ๐บ๐œŒ(๐‘ฅ,๐‘ฆ) of ๐‘…๐œŒ can be written as Now, since the variables separate pairwise, we haveThe formula for ๐‘’โˆ’๐ฟ๐‘—๐‘ก is given in [13]. It readsHence the formula for ๐บ๐œŒ(๐‘ฅ,๐‘ฆ) becomeswhereAn expansion of ๐ผ(๐‘ ) shows that from which (2.12) follows.

3. The Exterior Landau-Neumann Hamiltonian in โ„2๐‘‘

Let ๐พโŠ‚โ„2๐‘‘ be a simply connected compact domain with smooth boundary ฮ“ and let ฮฉ=โ„2๐‘‘โงต๐พ. We define the exterior Landau-Neumann Hamiltonian ๐ฟฮฉ in โ„‹ฮฉ=๐ฟ2(ฮฉ) by with magnetic Neumann boundary conditions Here ๐œˆ denotes the exterior normal to ฮ“. Our aim is to study how much the spectrum of ๐ฟฮฉ differs from the Landau levels discussed in the previous section. The first theorem below states that the eigenvalues of ๐ฟฮฉ 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 ๐œ‡โˆˆโ„• and each ๐œ€, 0<๐œ€<๐‘‘๐ต, the number of eigenvalues of ๐ฟฮฉ in the interval (ฮ›๐œ‡,ฮ›๐œ‡+๐œ€) is finite.

Denote by ๐‘™1(๐œ‡)โ‰ค๐‘™2(๐œ‡)โ‰คโ‹ฏ the eigenvalues of ๐ฟฮฉ in the interval (ฮ›๐œ‡โˆ’1,ฮ›๐œ‡) and by ๐‘(๐‘Ž,๐‘,๐‘‡) the number of eigenvalues of the operator ๐‘‡ in the interval (๐‘Ž,๐‘), counting multiplicities. Also, let Cap(๐พ) denote the logarithmic capacity of ๐พ; see [14, Chapter 2].Theorem 3.2. Let ๐œ‡โˆˆโ„•. (a)If ๐‘‘=1 then lim๐‘—โ†’โˆž(๐‘—!(ฮ›๐œ‡โˆ’๐‘™๐‘—(๐œ‡)))1/๐‘—=(๐ต/2)(Cap(๐พ))2.(b)If ๐‘‘>1 then ๐‘(ฮ›๐œ‡โˆ’1,ฮ›๐œ‡โˆ’๐œ†,๐ฟฮฉ)โˆผ(๐œ‡+๐‘‘โˆ’1๐‘‘โˆ’1)(1/๐‘‘!)(|ln๐œ†|/ln|ln๐œ†|)๐‘‘ as ๐œ†โ†˜0.

3.1. Proof of The Theorems

We want to compare the spectrum of the operators ๐ฟ and ๐ฟฮฉ. However, the expression ๐ฟโˆ’๐ฟฮฉ has no meaning since ๐ฟ and ๐ฟฮฉ act in different Hilbert spaces. We introduce the Hilbert space โ„‹๐พ=๐ฟ2(๐พ) and define the interior Landau-Neumann Hamiltonian ๐ฟ๐พ in โ„‹๐พ by the same formulas as in (3.1) and (3.2) but with ฮฉ replaced by ๐พ. We note that โ„‹=โ„‹๐พโŠ•โ„‹ฮฉ and define ๎‚๐ฟ asThe inverse of ๐ฟ๐พ is compact, so ๐ฟ๐พ has at most a finite number of eigenvalues in each interval (ฮ›๐œ‡โˆ’1,ฮ›๐œ‡). The operators ๐ฟ๐พ and ๐ฟฮฉ act in orthogonal subspaces of โ„‹, so ๎‚๐œŽ(๐ฟ)=๐œŽ(๐ฟ๐พ)โˆช๐œŽ(๐ฟฮฉ). This means that ๎‚๐ฟ has the same spectral asymptotics as ๐ฟฮฉ in each interval (ฮ›๐œ‡โˆ’1,ฮ›๐œ‡), so it is enough to prove the statements in Theorems 3.1 and 3.2 for the operator ๎‚๐ฟ instead of ๐ฟฮฉ.

Since the unbounded operators ๐ฟ and ๎‚๐ฟ have different domains, we cannot compare them directly. However, they act in the same Hilbert space, so we can compare their inverses. Letand setLemma 3.3. ๐‘‰ is nonnegative and compact.

Proof. See Section 4.1.

By Weyl's theorem, the essential spectrum of ๐‘… and ๎‚๐‘… coincides. Since ๎‚๐‘…=๐‘…+๐‘‰ and ๐‘‰โ‰ฅ0, Theorem 3.1 follows immediately from [15, Theorem 9.4.7] and the fact that ๐œŽ(๐‘…)=๐œŽess(๐‘…)={ฮ›๐œ‡โˆ’1}. We continue with the proof of Theorem 3.2.

Let ๐œ>0 be such that ((ฮ›๐œ‡โˆ’1โˆ’2๐œ,ฮ›๐œ‡โˆ’1+2๐œ)โงต{ฮ›๐œ‡โˆ’1})โˆฉ๐œŽess(๐‘…)=โˆ…. Denote the eigenvalues of ๐‘‡๐œ‡ byand the eigenvalues of ๎‚๐‘… in the interval (ฮ›๐œ‡โˆ’1,ฮ›๐œ‡โˆ’1+๐œ) byLemma 3.4. Given ๐œ€>0, there exists an integer ๐‘™ such that

Proof. See [4, Proposition 2.2].

Hence the study of the asymptotics of the eigenvalues of ๎‚๐‘… is reduced to the study of the eigenvalues of the Toeplitz-type operator ๐‘‡๐œ‡. For a bounded simply connected set ๐‘ˆ in โ„2๐‘‘, we define the Toeplitz operator ๐‘†๐‘ˆ๐œ‡ aswhere ๐œ’๐‘ˆ denotes the characteristic function of ๐‘ˆ. The following lemma reduces our problem to the study of these Toeplitz operators, which are easier to study than ๐‘‡๐œ‡.Lemma 3.5. Let ๐พ0โ‹๐พโ‹๐พ1 be compact domains such that ๐œ•๐พ๐‘–โˆฉฮ“=โˆ…. There exist a constant ๐ถ>0 and a subspace ๐’ฎโŠ‚โ„‹ of finite codimension such that for all ๐‘“โˆˆ๐’ฎ.

Proof. See Section 4.2.

The asymptotic expansion of the spectrum of ๐‘†๐‘ˆ๐œ‡ is given in the following lemma. Lemma 3.6. Denote by ๐‘ 1(๐œ‡)โ‰ฅ๐‘ 2(๐œ‡)โ‰ฅโ‹ฏ the eigenvalues of ๐‘†๐‘ˆ๐œ‡ and by ๐‘›(๐œ†,๐‘†๐‘ˆ๐œ‡) the number of eigenvalues of ๐‘†๐‘ˆ๐œ‡ greater than ๐œ† (counting multiplicity). Then
(a)if ๐‘‘=1 we have lim๐‘—โ†’โˆž(๐‘—!๐‘ ๐‘—(๐œ‡))1/๐‘—=(๐ต/2)(Cap(๐‘ˆ))2,(b) if ๐‘‘>1 we have ๐‘›(๐œ†,๐‘†๐‘ˆ๐œ‡)โˆผ(๐œ‡+๐‘‘โˆ’1๐‘‘โˆ’1)(1/๐‘‘!)(|ln๐œ†|/ln|ln๐œ†|)๐‘‘ as ๐œ†โ†˜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 ๐พ0 and ๐พ1 in Lemma 3.5 get closer and closer to our compact ๐พ we see that the eigenvalues {๐‘ก๐‘—(๐œ‡)} of ๐‘‡๐œ‡ satisfyif ๐‘‘=1, and if ๐‘‘>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 {๐‘Ÿ๐‘—(๐œ‡)}๎‚๐‘… satisfyif ๐‘‘=1, and
If we translate this in terms of ๎‚๐ฟ we getfor ๐‘‘=1, and for ๐‘‘>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 3.3

The operators ๐ฟ and ๎‚๐ฟ are defined by the same expression, but the domain of ๎‚๐ฟ is contained in the domain of ๐ฟ. It follows from [4, Proposition 2.1] that ๎‚๐ฟโˆ’๐ฟโ‰ฅ0, and hence ๎‚๐‘‰=๐‘…โˆ’๐‘…โ‰ฅ0.

Next we prove the compactness of ๐‘‰. Let ๐‘“ and ๐‘” belong to โ„‹. Also, let ๐‘ข=๐‘…๐‘“ and ๎‚๐‘ฃ=๐‘…๐‘”. Then ๐‘ข belongs to the domain of ๐ฟ and ๐‘ฃ belongs to the domain of ๎‚๐ฟ, so ๐‘ฃ=๐‘ฃ๐พโŠ•๐‘ฃฮฉ, and ๐ฟ๐พ๐‘ฃ๐พโŠ•๐ฟฮฉ๐‘ฃฮฉ=๐‘”. Integrating by parts and using (3.2) for ๐‘ฃ๐พ and ๐‘ฃฮฉ, we getHere d๐‘† denotes the surface measure on ฮ“.

Take a smooth cutoff function ๐œ’โˆˆ๐ถโˆž0(โ„2๐‘‘) such that ๐œ’(๐‘ฅ)=1 in a neighborhood of ๐พ. Then we can replace ๐‘ข and ๐‘ฃ by ฬƒ๐‘ข=๐œ’๐‘ข and ฬƒ๐‘ฃ=๐œ’๐‘ฃ in the right-hand side of (4.1). By local elliptic regularity we have that ฬƒ๐‘ขโˆˆ๐ป2(โ„2๐‘‘) and ฬƒ๐‘ฃโˆˆ๐ป2(โ„2๐‘‘โงตฮ“). However, the operator ฬƒ๐‘ขโ†ฆ๐œ•๐‘ฬƒ๐‘ข|ฮ“ is compact as considered from ๐ป2(โ„2๐‘‘) to ๐ฟ2(ฮ“) and both ฬƒ๎‚‹๐‘ฃ๐‘ฃโ†ฆฮฉ|ฮ“ and ฬƒ๎‚‹๐‘ฃ๐‘ฃโ†ฆ๐พ|ฮ“ are compact as considered from ๐ป2(โ„2๐‘‘โงตฮ“) to ๐ฟ2(ฮ“), so it follows that ๐‘‰ is compact.

4.2. Proof of Lemma 3.5

We start by showing that ๐‘‡๐œ‡, originally defined in ๐ฟ2(โ„2๐‘‘), can be reduced to an operator in ๐ฟ2(ฮ“). More precisely we show that ๐‘‡๐œ‡ can be realized as an elliptic pseudodifferential operator of order 1 on some subspace of ๐ฟ2(ฮ“) of finite codimension, and hence there exists a constant ๐ถ>0 such thatfor all ๐‘“ in that subspace.

Let ๐‘“ and ๐‘” belong to โ„‹. Also, let ๐‘ข=๐‘…๐‘“, ๎‚๐‘ฃ=๐‘…๐‘”, and ๐‘ค=๐‘…๐‘”. We saw in (4.1) that

To go further we will introduce the Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. Let ๐บ๐œŒ(๐‘ฅ,๐‘ฆ) be as in (2.16). We start with the single- and double-layer integral operators, defined byThe last two operators are compact on ๐ฟ2(ฮ“), since, by Lemma 2.1, their kernels have weak singularities. Moreover, since the kernel ๐บ0 has the same singularity as the Green kernel for the Laplace operator in โ„2๐‘‘ (see [18, Chapter 7, Section 11]), we have the following limit relations on ฮ“Using a Green-type formula for ๐ฟ in ๐พ we see thatIf we combine this with the limit relations (4.5), we get

A similar calculation for ฮฉ gives

It seems natural to do the following definitions.Definition 4.1. We define the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators in ๐พ and ฮฉ asRemark 4.2. The inverses above exist at least on a space of finite codimension. This follows from the fact that ๐ด is elliptic and ๐ต is compact.Lemma 4.3. The operator (๐‘๐ท)๐พโˆ’(๐‘๐ท)ฮฉ is an elliptic pseudodifferential operator of order โˆ’1.

Proof. Using a resolvent identity, we see that It follows from the asymptotic expansion of ๐บ0(๐‘ฅ,๐‘ฆ) in Lemma 2.1 and the fact that ๐บ0 is a Schwartz kernel (again, see [18, Chapter 7, Section 11]) that ๐ด is an elliptic pseudodifferential operator of order โˆ’1. Moreover the operator ๐ต 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 ๐‘‰. We haveSince we are interested in ๐‘‡๐œ‡ and not ๐‘‰, we may assume that ๐‘“ and ๐‘” belong to โ„’ฮ›๐œ‡. Then ๐‘ข=๐‘…๐‘“=ฮ›๐œ‡โˆ’1๐‘“ and ๐‘ค=๐‘…๐‘”=ฮ›๐œ‡โˆ’1๐‘”. For such ๐‘“ and ๐‘” we getor with the introduced operators aboveMoreover, (๐ท๐‘)๐พ is an elliptic pseudodifferential operator of order 1. This follows from the identity ๐ด(๐ท๐‘)๐พ=๐ตโˆ’(1/2)๐ผ, and the fact that ๐ด is an elliptic pseudodifferential operator of order โˆ’1. It follows from (4.13) that ๐‘‡๐œ‡ 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 ๐‘“ in โ„’ฮ›๐œ‡.

The lower bound. We prove that there exists a subspace ๎‚๐’ฎโŠ‚โ„’ฮ›๐œ‡ of finite codimension such that the lower bound in (3.10) is valid for all ๎‚๐’ฎ๐‘“โˆˆ. Since ๐‘“โˆˆโ„’ฮ›๐œ‡ we have ๐ฟ๐œ‡๐‘“โˆถ=(๐ฟโˆ’ฮ›๐œ‡)๐‘“=0 so ๐‘“ belongs to the kernel of the second-order elliptic operator ๐ฟ๐œ‡. Let ๐œ‘=๐‘“|ฮ“. We study the problemLet ๐ธ(๐‘ฅ,๐‘ฆ) be the Schwartz kernel for ๐ฟ๐œ‡. It is smooth away from the diagonal ๐‘ฅ=๐‘ฆ. One can repeat the theory with the single- and double-layer potentials for ๐ฟ๐œ‡ and write the solution ๐‘“ in the case it exists.

Let ๐ต๐œ‡ be the double-layer operator evaluated at the boundary,The operator ๐ต๐œ‡ is compact, since the kernel ๐œ•๐‘๐‘ฆ๐ธ(๐‘ฅ,๐‘ฆ) has a weak singularity at the diagonal ๐‘ฅ=๐‘ฆ. Thus there exists a subspace ๐’ฎ1โŠ‚๐ฟ2(ฮ“) of finite codimension such that the operator (1/2)๐ผ+๐ต๐œ‡ is invertible on ๐’ฎ1. Hence, there exists a subspace ๎‚๐’ฎโŠ‚โ„’ฮ›๐œ‡ of finite codimension where we have the representation formulafor all ๎‚๐’ฎ๐‘“โˆˆ. The inequality โ€–๐‘“โ€–๐ฟ2(๐พ0)โ‰ค๐ถโ€–๐‘“โ€–๐ฟ2(ฮ“) follows easily from (4.16) for all such functions ๐‘“.

Since we also have โ€–๐‘“โ€–๐ฟ2(ฮ“)โ‰ค๐ถโ€–๐‘“โ€–๐ป1(ฮ“) 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 inequalitiesHowever, the first inequality is just the Trace theorem, the second is the Sobolev-Rellich embedding theorem. We note that ๐ฟ๐œ‡๐‘“=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 ๐พ is a Reinhardt domain one can strengthen part (๐‘) of Lemma 3.6. Assume that ๐พโˆ˜, the interior of ๐พ, is a Reinhardt domain. This means that 0โˆˆ๐พโˆ˜ and if ๐‘งโˆˆ๐พโˆ˜, then the setis a subset of ๐พโˆ˜. If the setis convex in the usual sense, then ๐พโˆ˜ is said to be logarithmically convex, and ๐พโˆ˜ is a domain of holomorphy. Denote by ๐‘‰๐พโˆถโ„๐‘‘โ†’โ„ the function defined byWe denote by ๐ฝโˆถโ„ฑ2๐ตโ†’๎‚‹โ„‹โˆถ=๐ฟ2(๐พ,๐‘’โˆ’(๐ต/2)|๐‘ง|2d๐‘š(๐‘ง)) the embedding operator. The ๐‘  -values ๐‘ ๎๐œ…, ๎๐œ…โˆˆโ„•๐‘‘, of ๐ฝ coincide with the numbers(we remind the reader of the notation of eigenvalues in Lemma 3.6). Unlike the case ๐‘‘=1, see [10], it is natural to numerate the eigenvalues by the ๐‘‘-tuples ๎๐œ…=(๐œ…1,โ€ฆ,๐œ…๐‘‘), just as for the eigenvalues of the Laplace operator in the unit cube [0,1]๐‘‘, where the eigenvalues are given by (2๐œ‹)โˆ’๐‘‘|๎๐œ…|22=(2๐œ‹)โˆ’๐‘‘(๐œ…21+โ‹ฏ+๐œ…2๐‘‘).Lemma 5.1. Let ๐‘‘>1 and ๐œ”=๎๐œ…/|๎๐œ…|. Then

Proof. The denominator in (5.4) is easily calculated to be For the numerator, we do estimations from above and below, as in [12]. First, note thatwhere d๎‚๐‘š(๐‘ฅ) is the transformed measure. It is clear thatFor the inequality in the other direction, fix ๐›ฟ>0. The hyperplanecuts log|๐พ| in two components. Let ๐‘ƒ๐›ฟ be the component for which the inequality โŸจ๎๐œ…,๐‘ฅโŸฉโ‰ฅ(1โˆ’๐›ฟ)๐‘‰๐พ(๎๐œ…) holds. Then we havewhere ๐ถ๐›ฟ=โˆซ๐‘ƒ๐›ฟd๎‚๐‘š(๐‘ฅ)>0. It follows thatfrom 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.

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