Advances in Mathematical Physics

Advances in Mathematical Physics / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 873704 | https://doi.org/10.1155/2009/873704

Mikael Persson, "Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian", Advances in Mathematical Physics, vol. 2009, Article ID 873704, 15 pages, 2009. https://doi.org/10.1155/2009/873704

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

Academic Editor: Pavel Exner
Received27 Jun 2008
Accepted22 Oct 2008
Published25 Nov 2008

Abstract

We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in 2𝑑, 𝑑1. 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 [411]. 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 (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 2𝑑

We denote by 𝑥=(𝑥1,,𝑥2𝑑) a point in 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 2𝑑 describes a charged, spinless particle in this homogeneous magnetic field. It is given byand is essentially self-adjoint on the set 𝐶0(2𝑑) in the usual Hilbert space =𝐿2(2𝑑). For 𝑗=1,,𝑑, we also introduce the self-adjoint operatorsin the Hilbert spaces 𝑗=𝐿2(2). Note that =𝑑𝑗=1𝑗, and

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={0,1,2,}, each of infinite multiplicity. Let 𝜅=(𝜅1,𝜅𝑑)𝑑 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 eigenvaluesNote that Λ𝜅=Λ𝜅 if |𝜅|=|𝜅|. Hence the spectrum of 𝐿 consists of eigenvalues of the form Λ𝜇=𝐵(2𝜇+𝑑), 𝜇.

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)𝑑, where 𝑧𝑗=𝑥2𝑗1+𝑖𝑥2𝑗. Also, we use the scalar potential 𝑊(𝑧)=(𝐵/4)|𝑧|2 and the complex derivatives

We define creation and annihilation operators 𝒬𝑗, 𝒬𝑗 asand note that The notation 𝒬𝑗 for the creation operators is motivated by the fact that it is the formal adjoint of 𝒬𝑗 in .

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=𝑒𝑊𝑢 is an entire function, so via multiplication by 𝑒𝑊 the eigenspace 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 Λ𝜇 of the Landau level Λ𝜇 if and only if it can be written in the formwhere (𝒬)𝜅=(𝒬1)𝜅1(𝒬𝑑)𝜅𝑑 and 𝑓𝜅 all belong to 2𝐵. The multiplicity of each eigenvalue Λ𝜇 is infinite. We denote by 𝒫Λ𝜅 and 𝒫Λ𝜇 the projection onto the eigenspaces Λ𝜅 and Λ𝜇, respectively, and note by (2.16) that the orthogonal decompositionshold in .

2.3. The Resolvent

Let 𝑅𝜌=(𝐿+𝜌𝐼)1 be the resolvent of 𝐿, 𝜌0. 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: as |𝑥𝑦|0.

Proof. The kernel 𝐺𝜌(𝑥,𝑦) of 𝑅𝜌 can be written as Now, since the variables separate pairwise, we haveThe formula for 𝑒𝐿𝑗𝑡 is given in [13]. It readsHence the formula for 𝐺𝜌(𝑥,𝑦) becomeswhereAn expansion of 𝐼(𝑠) shows that from which (2.12) follows.

3. The Exterior Landau-Neumann Hamiltonian in 2𝑑

Let 𝐾2𝑑 be a simply connected compact domain with smooth boundary Γ and let Ω=2𝑑𝐾. We define the exterior Landau-Neumann Hamiltonian 𝐿Ω in 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 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 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 (Λ𝜇1,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. (a)If 𝑑=1 then lim𝑗(𝑗!(Λ𝜇𝑙𝑗(𝜇)))1/𝑗=(𝐵/2)(Cap(𝐾))2.(b)If 𝑑>1 then 𝑁(Λ𝜇1,Λ𝜇𝜆,𝐿Ω)(𝜇+𝑑1𝑑1)(1/𝑑!)(|ln𝜆|/ln|ln𝜆|)𝑑 as 𝜆0.

3.1. Proof of The Theorems

We want to compare the spectrum of the operators 𝐿 and 𝐿Ω. However, the expression 𝐿𝐿Ω has no meaning since 𝐿 and 𝐿Ω act in different Hilbert spaces. We introduce the Hilbert space 𝐾=𝐿2(𝐾) and define the interior Landau-Neumann Hamiltonian 𝐿𝐾 in 𝐾 by the same formulas as in (3.1) and (3.2) but with Ω replaced by 𝐾. We note that =𝐾Ω and define 𝐿 asThe inverse of 𝐿𝐾 is compact, so 𝐿𝐾 has at most a finite number of eigenvalues in each interval (Λ𝜇1,Λ𝜇). The operators 𝐿𝐾 and 𝐿Ω act in orthogonal subspaces of , so 𝜎(𝐿)=𝜎(𝐿𝐾)𝜎(𝐿Ω). This means that 𝐿 has the same spectral asymptotics as 𝐿Ω in each interval (Λ𝜇1,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. 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 𝑉0, Theorem 3.1 follows immediately from [15, Theorem 9.4.7] and the fact that 𝜎(𝑅)=𝜎ess(𝑅)={Λ𝜇1}. We continue with the proof of Theorem 3.2.

Let 𝜏>0 be such that ((Λ𝜇12𝜏,Λ𝜇1+2𝜏){Λ𝜇1})𝜎ess(𝑅)=. Denote the eigenvalues of 𝑇𝜇 byand the eigenvalues of 𝑅 in the interval (Λ𝜇1,Λ𝜇1+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 𝑇𝜇. For a bounded simply connected set 𝑈 in 2𝑑, we define the Toeplitz operator 𝑆𝑈𝜇 aswhere 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 𝐾0𝐾𝐾1 be compact domains such that 𝜕𝐾𝑖Γ=. There exist a constant 𝐶>0 and a subspace 𝒮 of finite codimension such that for all 𝑓𝒮.

Proof. See Section 4.2.

The asymptotic expansion of the spectrum of 𝑆𝑈𝜇 is given in the following lemma. Lemma 3.6. Denote by 𝑠1(𝜇)𝑠2(𝜇) the eigenvalues of 𝑆𝑈𝜇 and by 𝑛(𝜆,𝑆𝑈𝜇) the number of eigenvalues of 𝑆𝑈𝜇 greater than 𝜆 (counting multiplicity). Then
(a)if 𝑑=1 we have lim𝑗(𝑗!𝑠𝑗(𝜇))1/𝑗=(𝐵/2)(Cap(𝑈))2,(b) if 𝑑>1 we have 𝑛(𝜆,𝑆𝑈𝜇)(𝜇+𝑑1𝑑1)(1/𝑑!)(|ln𝜆|/ln|ln𝜆|)𝑑 as 𝜆0.

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 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 {𝑟𝑗(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 𝐿𝐿0, and hence 𝑉=𝑅𝑅0.

Next we prove the compactness of 𝑉. Let 𝑓 and 𝑔 belong to . Also, let 𝑢=𝑅𝑓 and 𝑣=𝑅𝑔. Then 𝑢 belongs to the domain of 𝐿 and 𝑣 belongs to the domain of 𝐿, so 𝑣=𝑣𝐾𝑣Ω, and 𝐿𝐾𝑣𝐾𝐿Ω𝑣Ω=𝑔. Integrating by parts and using (3.2) for 𝑣𝐾 and 𝑣Ω, we getHere d𝑆 denotes the surface measure on Γ.

Take a smooth cutoff function 𝜒𝐶0(2𝑑) 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 ̃𝑢𝐻2(2𝑑) and ̃𝑣𝐻2(2𝑑Γ). However, the operator ̃𝑢𝜕𝑁̃𝑢|Γ is compact as considered from 𝐻2(2𝑑) to 𝐿2(Γ) and both ̃𝑣𝑣Ω|Γ and ̃𝑣𝑣𝐾|Γ are compact as considered from 𝐻2(2𝑑Γ) to 𝐿2(Γ), so it follows that 𝑉 is compact.

4.2. Proof of Lemma 3.5

We start by showing that 𝑇𝜇, originally defined in 𝐿2(2𝑑), 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 thatfor all 𝑓 in that subspace.

Let 𝑓 and 𝑔 belong to . Also, let 𝑢=𝑅𝑓, 𝑣=𝑅𝑔, 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 𝐺𝜌(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 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 𝑇𝜇 and not 𝑉, we may assume that 𝑓 and 𝑔 belong to Λ𝜇. Then 𝑢=𝑅𝑓=Λ𝜇1𝑓 and 𝑤=𝑅𝑔=Λ𝜇1𝑔. 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 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 Λ𝜇.

The lower bound. We prove that there exists a subspace 𝒮Λ𝜇 of finite codimension such that the lower bound in (3.10) is valid for all 𝒮𝑓. Since 𝑓Λ𝜇 we have 𝐿𝜇𝑓=(𝐿Λ𝜇)𝑓=0 so 𝑓 belongs to the kernel of the second-order elliptic operator 𝐿𝜇. Let 𝜑=𝑓|Γ. We study the problemLet 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,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 𝒮1𝐿2(Γ) of finite codimension such that the operator (1/2)𝐼+𝐵𝜇 is invertible on 𝒮1. Hence, there exists a subspace 𝒮Λ𝜇 of finite codimension where we have the representation formulafor all 𝒮𝑓. The inequality 𝑓𝐿2(𝐾0)𝐶𝑓𝐿2(Γ) follows easily from (4.16) for all such functions 𝑓.

Since we also have 𝑓𝐿2(Γ)𝐶𝑓𝐻1(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 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 0𝐾 and if 𝑧𝐾, 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 𝑉𝐾𝑑 the function defined byWe denote by 𝐽2𝐵=𝐿2(𝐾,𝑒(𝐵/2)|𝑧|2d𝑚(𝑧)) the embedding operator. The 𝑠 -values 𝑠𝜅, 𝜅𝑑, 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,,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(𝜅21++𝜅2𝑑).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 𝑃𝛿 be the component for which the inequality 𝜅,𝑥(1𝛿)𝑉𝐾(𝜅) holds. Then we havewhere 𝐶𝛿=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.

References

  1. V. Fock, “Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld,” Zeitschrift für Physik A, vol. 47, no. 5-6, pp. 446–448, 1928. View at: Publisher Site | Google Scholar
  2. L. Landau, “Diamagnetismus der Metalle,” Zeitschrift für Physik A, vol. 64, no. 9-10, pp. 629–637, 1930. View at: Publisher Site | Google Scholar
  3. K. Hornberger and U. Smilansky, “Magnetic edge states,” Physics Reports, vol. 367, no. 4, pp. 249–385, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  4. A. Pushnitski and G. Rozenblum, “Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain,” Documenta Mathematica, vol. 12, pp. 569–586, 2007. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  5. G. D. Raĭkov, “Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips,” Communications in Partial Differential Equations, vol. 15, no. 3, pp. 407–434, 1990. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. G. D. Raikov and S. Warzel, “Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentials,” Comptes Rendus Mathematique, vol. 335, no. 8, pp. 683–688, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  7. G. D. Raikov and S. Warzel, “Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials,” Reviews in Mathematical Physics, vol. 14, no. 10, pp. 1051–1072, 2002. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  8. M. Melgaard and G. Rozenblum, “Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank,” Communications in Partial Differential Equations, vol. 28, no. 3-4, pp. 697–736, 2003. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  9. G. D. Raikov, “Spectral asymptotics for the perturbed 2D Pauli operator with oscillating magnetic fields. I: non-zero mean value of the magnetic field,” Markov Processes and Related Fields, vol. 9, no. 4, pp. 775–794, 2003. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  10. N. Filonov and A. Pushnitski, “Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains,” Communications in Mathematical Physics, vol. 264, no. 3, pp. 759–772, 2006. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  11. G. Rozenblum and A. V. Sobolev, “Discrete spectrum distribution of the landau operator perturbed by an expanding electric potential,” American Mathematical Society Translations, Series 2, vol. 255, pp. 169–190, 2008. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  12. O. G. Parfënov, “The singular values of the imbedding operators of some classes of analytic functions of several variables,” Journal of Mathematical Sciences, vol. 72, no. 6, pp. 3428–3434, 1994. View at: Publisher Site | Google Scholar | MathSciNet
  13. B. Simon, Functional Integration and Quantum Physics, vol. 86 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, 1979. View at: Zentralblatt MATH | MathSciNet
  14. N. S. Landkof, Foundations of Modern Potential Theory, vol. 180 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, 1972. View at: Zentralblatt MATH | MathSciNet
  15. M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Mathematics and Its Applications, D. Reidel, Dordrecht, Holland, 1987. View at: Zentralblatt MATH | MathSciNet
  16. M. E. Taylor, Pseudodifferential Operators, vol. 34 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1981. View at: Zentralblatt MATH | MathSciNet
  17. M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH, Berlin, Germany, 1999. View at: Zentralblatt MATH | MathSciNet
  18. M. E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, vol. 116 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1996. View at: Zentralblatt MATH | MathSciNet

Copyright © 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.

Related articles

No related content is available yet for this article.
 PDF Download Citation Citation
 Download other formatsMore
 Order printed copiesOrder
Views882
Downloads685
Citations

Related articles

No related content is available yet for this article.

Article of the Year Award: Outstanding research contributions of 2021, as selected by our Chief Editors. Read the winning articles.