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

Abstract

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in ℝ^2𝑑, 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 (ℝ^2𝑑, 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 ℝ^2𝑑

We denote by 𝑥=(𝑥¹,…,𝑥^2𝑑) a point in ℝ^2𝑑. Let 𝐵>0 and denote by ^→𝑎 the magnetic vector potentialIt corresponds to an isotropic magnetic field of constant strength 𝐵. The Landau Hamiltonian 𝐿 in ℝ^2𝑑 describes a charged, spinless particle in this homogeneous magnetic field. It is given byand is essentially self-adjoint on the set 𝐶^∞0(ℝ^2𝑑) in the usual Hilbert space ℋ=𝐿₂(ℝ^2𝑑). For 𝑗=1,…,𝑑, we also introduce the self-adjoint operatorsin the Hilbert spaces ℋ_𝑗=𝐿₂(ℝ²). Note that ℋ=⊗^𝑑𝑗=1ℋ_𝑗, and

2.1. Landau Levels

The spectrum of each two-dimensional Landau Hamiltonian 𝐿_𝑗 consists of the so-called Landau levels, eigenvalues 𝐵(2𝑞+1), 𝑞∈ℕ∶={0,1,2,…}, each of infinite multiplicity. Let 𝜅=(𝜅₁,…𝜅_𝑑)∈ℕ^𝑑 be a multi-index. We denote by |𝜅|=𝜅₁+⋯+𝜅_𝑑 the length of the multi-index 𝜅 and also set 𝜅!=𝜅₁!⋯𝜅_𝑑!. From (2.4) it follows that the spectrum of 𝐿 consists of the infinitely degenerate eigenvaluesNote that Λ𝜅=Λ𝜅^′ if |𝜅|=|𝜅^|. Hence the spectrum of 𝐿 consists of eigenvalues of the form Λ_𝜇=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,…,𝑧^𝑑)∈ℂ^𝑑, where 𝑧^𝑗=𝑥^2𝑗−1+𝑖𝑥^2𝑗. Also, we use the scalar potential 𝑊(𝑧)=−(𝐵/4)|𝑧|² and the complex derivatives

We define creation and annihilation operators 𝒬^∗𝑗, 𝒬_𝑗 asand note that The notation 𝒬^∗𝑗 for the creation operators is motivated by the fact that it is the formal adjoint of 𝒬_𝑗 in ℋ.

A function 𝑢 belongs to the lowest Landau level Λ₀ if and only if 𝒬_𝑗𝑢=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 ℒ_Λ0 corresponding to Λ₀ is equivalent to the Fock space

Here, and elsewhere, d𝑚 denotes the Lebesgue measure. A function 𝑢 belongs to the eigenspace ℒ_Λ𝜇 of the Landau level Λ_𝜇 if and only if it can be written in the formwhere (𝒬^∗)𝜅=(𝒬^∗1)^𝜅1⋯(𝒬^∗𝑑)^𝜅𝑑 and 𝑓𝜅 all belong to ℱ^2𝐵. The multiplicity of each eigenvalue Λ_𝜇 is infinite. We denote by 𝒫_Λ𝜅 and 𝒫_Λ𝜇 the projection onto the eigenspaces ℒ_Λ𝜅 and ℒ_Λ𝜇, respectively, and note by (2.16) that the orthogonal decompositionshold in ℋ.

2.3. The Resolvent

Let 𝑅_𝜌=(𝐿+𝜌𝐼)⁻¹ be the resolvent of 𝐿, 𝜌≥0. An explicit formula of the kernel 𝐺_𝜌(𝑥,𝑦) of 𝑅_𝜌 was given in [3] for 𝑑=1. In Section 4.2, we will use the behavior of 𝐺_𝜌(𝑥,𝑦) near the diagonal 𝑥=𝑦, given in the following lemma.Lemma 2.1. 𝑅_𝜌 is an integral operator with kernel 𝐺_𝜌(𝑥,𝑦) that has the following singularity at the diagonal: as |𝑥−𝑦|→0.

Proof. The kernel 𝐺_𝜌(𝑥,𝑦) of 𝑅_𝜌 can be written as Now, since the variables separate pairwise, we haveThe formula for 𝑒^{−𝐿𝑗𝑡} is given in [13]. It readsHence the formula for 𝐺_𝜌(𝑥,𝑦) becomeswhereAn expansion of 𝐼(𝑠) shows that from which (2.12) follows.

3. The Exterior Landau-Neumann Hamiltonian in ℝ^2𝑑

Let 𝐾⊂ℝ^2𝑑 be a simply connected compact domain with smooth boundary Γ and let Ω=ℝ^2𝑑⧵𝐾. We define the exterior Landau-Neumann Hamiltonian 𝐿_Ω in ℋ_Ω=𝐿₂(Ω) by with magnetic Neumann boundary conditions Here 𝜈 denotes the exterior normal to Γ. Our aim is to study how much the spectrum of 𝐿_Ω differs from the Landau levels discussed in the previous section. The first theorem below states that the eigenvalues of 𝐿_Ω 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 𝐿_Ω in the interval (Λ_𝜇,Λ_𝜇+𝜀) is finite.

Denote by 𝑙^1(𝜇)≤𝑙^2(𝜇)≤⋯ the eigenvalues of 𝐿_Ω in the interval (Λ_𝜇−1,Λ_𝜇) 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. (a)If 𝑑=1 then lim_𝑗→∞(𝑗!(Λ_𝜇−𝑙^𝑗(𝜇)))^1/𝑗=(𝐵/2)(Cap(𝐾))².(b)If 𝑑>1 then 𝑁(Λ_𝜇−1,Λ_𝜇−𝜆,𝐿_Ω)∼(^{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 𝐿_Ω. However, the expression 𝐿−𝐿_Ω has no meaning since 𝐿 and 𝐿_Ω act in different Hilbert spaces. We introduce the Hilbert space ℋ_𝐾=𝐿₂(𝐾) and define the interior Landau-Neumann Hamiltonian 𝐿_𝐾 in ℋ_𝐾 by the same formulas as in (3.1) and (3.2) but with Ω replaced by 𝐾. We note that ℋ=ℋ_𝐾⊕ℋ_Ω and define 𝐿 asThe inverse of 𝐿_𝐾 is compact, so 𝐿_𝐾 has at most a finite number of eigenvalues in each interval (Λ_𝜇−1,Λ_𝜇). The operators 𝐿_𝐾 and 𝐿_Ω act in orthogonal subspaces of ℋ, so 𝜎(𝐿)=𝜎(𝐿_𝐾)∪𝜎(𝐿_Ω). This means that 𝐿 has the same spectral asymptotics as 𝐿_Ω in each interval (Λ_𝜇−1,Λ_𝜇), so it is enough to prove the statements in Theorems 3.1 and 3.2 for the operator 𝐿 instead of 𝐿_Ω.

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. Letand setLemma 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)=𝜎_ess(𝑅)={Λ^𝜇−1}. We continue with the proof of Theorem 3.2.

Let 𝜏>0 be such that ((Λ^𝜇−1−2𝜏,Λ^𝜇−1+2𝜏)⧵{Λ^𝜇−1})∩𝜎_ess(𝑅)=∅. Denote the eigenvalues of 𝑇_𝜇 byand the eigenvalues of 𝑅 in the interval (Λ^𝜇−1,Λ^𝜇−1+𝜏) byLemma 3.4. Given 𝜀>0, there exists an integer 𝑙 such that

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 𝑇_𝜇. For a bounded simply connected set 𝑈 in ℝ^2𝑑, we define the Toeplitz operator 𝑆^𝑈𝜇 aswhere 𝜒_𝑈 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 𝑇_𝜇.Lemma 3.5. Let 𝐾₀⋐𝐾⋐𝐾₁ be compact domains such that 𝜕𝐾_𝑖∩Γ=∅. There exist a constant 𝐶>0 and a subspace 𝒮⊂ℋ of finite codimension such that for all 𝑓∈𝒮.

Proof. See Section 4.2.

The asymptotic expansion of the spectrum of 𝑆^𝑈𝜇 is given in the following lemma. Lemma 3.6. Denote by 𝑠^1(𝜇)≥𝑠^2(𝜇)≥⋯ the eigenvalues of 𝑆^𝑈𝜇 and by 𝑛(𝜆,𝑆^𝑈𝜇) the number of eigenvalues of 𝑆^𝑈𝜇 greater than 𝜆 (counting multiplicity). Then
(a)if 𝑑=1 we have lim_𝑗→∞(𝑗!𝑠^𝑗(𝜇))^1/𝑗=(𝐵/2)(Cap(𝑈))²,(b) if 𝑑>1 we have 𝑛(𝜆,𝑆^𝑈𝜇)∼(^{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 {𝑡^𝑗(𝜇)} of 𝑇_𝜇 satisfyif 𝑑=1, and 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 {𝑟^𝑗(𝜇)}𝑅 satisfyif 𝑑=1, and
If we translate this in terms of 𝐿 we getfor 𝑑=1, and 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 3.3

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=𝑣_𝐾⊕𝑣_Ω, and 𝐿_𝐾𝑣_𝐾⊕𝐿_Ω𝑣_Ω=𝑔. Integrating by parts and using (3.2) for 𝑣_𝐾 and 𝑣_Ω, we getHere d𝑆 denotes the surface measure on Γ.

Take a smooth cutoff function 𝜒∈𝐶^∞0(ℝ^2𝑑) such that 𝜒(𝑥)=1 in a neighborhood of 𝐾. Then we can replace 𝑢 and 𝑣 by 𝑢̃=𝜒𝑢 and 𝑣̃=𝜒𝑣 in the right-hand side of (4.1). By local elliptic regularity we have that 𝑢̃∈𝐻²(ℝ^2𝑑) and 𝑣̃∈𝐻²(ℝ^2𝑑⧵Γ). However, the operator 𝑢̃↦𝜕_𝑁𝑢|̃_Γ is compact as considered from 𝐻²(ℝ^2𝑑) to 𝐿₂(Γ) and both 𝑣̃↦𝑣_Ω|_Γ and 𝑣̃↦𝑣_𝐾|_Γ are compact as considered from 𝐻²(ℝ^2𝑑⧵Γ) to 𝐿₂(Γ), so it follows that 𝑉 is compact.

4.2. Proof of Lemma 3.5

We start by showing that 𝑇_𝜇, originally defined in 𝐿₂(ℝ^2𝑑), can be reduced to an operator in 𝐿₂(Γ). More precisely we show that 𝑇_𝜇 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 thatfor 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

To go further we will introduce the Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. Let 𝐺_𝜌(𝑥,𝑦) be as in (2.16). We start with the single- and double-layer integral operators, defined byThe 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 ℝ^2𝑑 (see [18, Chapter 7, Section 11]), we have the following limit relations on ΓUsing a Green-type formula for 𝐿 in 𝐾 we see thatIf we combine this with the limit relations (4.5), we get

A similar calculation for Ω gives

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 asRemark 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 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 haveSince we are interested in 𝑇_𝜇 and not 𝑉, we may assume that 𝑓 and 𝑔 belong to ℒ_Λ𝜇. Then 𝑢=𝑅𝑓=Λ^𝜇−1𝑓 and 𝑤=𝑅𝑔=Λ^𝜇−1𝑔. For such 𝑓 and 𝑔 we getor with the introduced operators aboveMoreover, (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 𝑇_𝜇 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 ℒ_Λ𝜇.

The lower bound. We prove that there exists a subspace 𝒮⊂ℒ_Λ𝜇 of finite codimension such that the lower bound in (3.10) is valid for all 𝑓∈𝒮. Since 𝑓∈ℒ_Λ𝜇 we have 𝐿_𝜇𝑓∶=(𝐿−Λ_𝜇)𝑓=0 so 𝑓 belongs to the kernel of the second-order elliptic operator 𝐿_𝜇. Let 𝜑=𝑓|_Γ. We study the problemLet 𝐸(𝑥,𝑦) be the Schwartz kernel for 𝐿_𝜇. It is smooth away from the diagonal 𝑥=𝑦. One can repeat the theory with the single- and double-layer potentials for 𝐿_𝜇 and write the solution 𝑓 in the case it exists.

Let 𝐵_𝜇 be the double-layer operator evaluated at the boundary,The operator 𝐵_𝜇 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+𝐵_𝜇 is invertible on 𝒮₁. Hence, there exists a subspace 𝒮⊂ℒ_Λ𝜇 of finite codimension where we have the representation formulafor 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 inequalitiesHowever, the first inequality is just the Trace theorem, the second is the Sobolev-Rellich embedding theorem. We note that 𝐿_𝜇𝑓=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 setis a subset of 𝐾^∘. If the setis convex in the usual sense, then 𝐾^∘ is said to be logarithmically convex, and 𝐾^∘ is a domain of holomorphy. Denote by 𝑉_𝐾∶ℝ^𝑑→ℝ the function defined byWe denote by 𝐽∶ℱ^2𝐵→ℋ∶=𝐿₂(𝐾,𝑒^{−(𝐵/2)|𝑧|2}d𝑚(𝑧)) the embedding operator. The 𝑠 -values 𝑠𝜅, 𝜅∈ℕ^𝑑, of 𝐽 coincide with the numbers(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,…,𝜅_𝑑), 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(𝜅²¹+⋯+𝜅^2𝑑).Lemma 5.1. Let 𝑑>1 and 𝜔=𝜅/|𝜅|. 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 d𝑚(𝑥) is the transformed measure. It is clear thatFor the inequality in the other direction, fix 𝛿>0. The hyperplanecuts log|𝐾| in two components. Let 𝑃_𝛿 be the component for which the inequality ⟨𝜅,𝑥⟩≥(1−𝛿)𝑉_𝐾(𝜅) holds. Then we havewhere 𝐶_𝛿=∫_𝑃𝛿d𝑚(𝑥)>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.