Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian
We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in , . 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.
The Landau Hamiltonian describes a charged particle moving in a plane, influenced by a constant magnetic field of strength orthogonal to the plane. It is a classical result, see [1, 2], that the spectrum of the Landau Hamiltonian consists of infinitely degenerate eigenvalues , , 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 , 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 , 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 .
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 (, ), we also consider the case when the compact obstacle is a Reinhardt domain. We use some ideas from  to prove a more precise asymptotic formula for the eigenvalues. This is done in Section 5.
2. The Landau Hamiltonian in
We denote by a point in . Let and denote by the magnetic vector potentialIt corresponds to an isotropic magnetic field of constant strength . The Landau Hamiltonian in describes a charged, spinless particle in this homogeneous magnetic field. It is given byand is essentially self-adjoint on the set in the usual Hilbert space . For we also introduce the self-adjoint operatorsin the Hilbert spaces . Note that , and
2.1. Landau Levels
The spectrum of each two-dimensional Landau Hamiltonian consists of the so-called Landau levels, eigenvalues , , each of infinite multiplicity. Let be a multi-index. We denote by the length of the multi-index and also set . From (2.4) it follows that the spectrum of consists of the infinitely degenerate eigenvaluesNote that if . Hence the spectrum of consists of eigenvalues of the form , .
2.2. Creation and Annihilation Operators
The structure of the eigenspaces of has been described before in . We give the results without proofs. It is convenient to introduce complex notation. Let , where . Also, we use the scalar potential and the complex derivatives
We define creation and annihilation operators , asand note that The notation for the creation operators is motivated by the fact that it is the formal adjoint of in .
A function belongs to the lowest Landau level if and only if for . This means that the function is an entire function, so via multiplication by the eigenspace corresponding to is equivalent to the Fock space
Here, and elsewhere, denotes the Lebesgue measure. A function belongs to the eigenspace of the Landau level if and only if it can be written in the formwhere and all belong to . The multiplicity of each eigenvalue is infinite. We denote by and the projection onto the eigenspaces and , respectively, and note by (2.16) that the orthogonal decompositionshold in .
2.3. The Resolvent
Let be the resolvent of , . An explicit formula of the kernel of was given in  for . In Section 4.2, we will use the behavior of near the diagonal , given in the following lemma.Lemma 2.1. is an integral operator with kernel that has the following singularity at the diagonal: as .
Proof. The kernel of can be written as Now, since the variables separate pairwise, we haveThe formula for is given in . It readsHence the formula for becomeswhereAn expansion of shows that from which (2.12) follows.
3. The Exterior Landau-Neumann Hamiltonian in
Let be a simply connected compact domain with smooth boundary and let . We define the exterior Landau-Neumann Hamiltonian in by with magnetic Neumann boundary conditions Here denotes the exterior normal to . Our aim is to study how much the spectrum of differs from the Landau levels discussed in the previous section. The first theorem below states that the eigenvalues of 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 and each , , the number of eigenvalues of in the interval is finite.
Denote by the eigenvalues of in the interval and by the number of eigenvalues of the operator in the interval , counting multiplicities. Also, let denote the logarithmic capacity of ; see [14, Chapter 2].Theorem 3.2. Let . (a)If then .(b)If then as .
3.1. Proof of The Theorems
We want to compare the spectrum of the operators and . However, the expression has no meaning since and act in different Hilbert spaces. We introduce the Hilbert space and define the interior Landau-Neumann Hamiltonian in by the same formulas as in (3.1) and (3.2) but with replaced by . We note that and define asThe inverse of is compact, so has at most a finite number of eigenvalues in each interval . The operators and act in orthogonal subspaces of , so . This means that has the same spectral asymptotics as in each interval , so it is enough to prove the statements in Theorems 3.1 and 3.2 for the operator instead of .
Since the unbounded operators and 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. is nonnegative and compact.
Proof. See Section 4.1.
Let be such that . Denote the eigenvalues of byand the eigenvalues of in the interval byLemma 3.4. Given there exists an integer such that
Proof. See [4, Proposition 2.2].
Hence the study of the asymptotics of the eigenvalues of is reduced to the study of the eigenvalues of the Toeplitz-type operator . For a bounded simply connected set in , we define the Toeplitz operator aswhere denotes the characteristic function of . The following lemma reduces our problem to the study of these Toeplitz operators, which are easier to study than .Lemma 3.5. Let be compact domains such that . There exist a constant and a subspace of finite codimension such that for all .
Proof. See Section 4.2.
The asymptotic expansion of the spectrum of is given in the following lemma.
Lemma 3.6. Denote by the eigenvalues of and by the number of eigenvalues of greater than (counting multiplicity). Then
(a)if we have ,(b) if we have as .
Proof. We are now able to finish the proof of Theorem
letting and in Lemma 3.5 get closer and closer to our
compact we see that the eigenvalues of satisfyif ,
and if .
Since neither formula
sensitive for finite shifts in the indices,
it follows from
that the eigenvalues of satisfyif ,
If we translate this in terms of we getfor , and for . 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  and [17, Chapter 5].
4.1. Proof of Lemma 3.3
The operators and are defined by the same expression, but the domain of is contained in the domain of . It follows from [4, Proposition 2.1] that , and hence .
Next we prove the compactness of . Let and belong to . Also, let and . Then belongs to the domain of and belongs to the domain of , so , and . Integrating by parts and using (3.2) for and , we getHere denotes the surface measure on .
Take a smooth cutoff function such that in a neighborhood of . Then we can replace and by and in the right-hand side of (4.1). By local elliptic regularity we have that and . However, the operator is compact as considered from to and both and are compact as considered from to , so it follows that is compact.
4.2. Proof of Lemma 3.5
We start by showing that , originally defined in , can be reduced to an operator in . More precisely we show that can be realized as an elliptic pseudodifferential operator of order 1 on some subspace of of finite codimension, and hence there exists a constant such thatfor all in that subspace.
Let and belong to . Also, let , and . We saw in (4.1) that
To go further we will introduce the Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. Let be as in (2.16). We start with the single- and double-layer integral operators, defined byThe last two operators are compact on , since, by Lemma 2.1, their kernels have weak singularities. Moreover, since the kernel has the same singularity as the Green kernel for the Laplace operator in (see [18, Chapter 7, Section 11]), we have the following limit relations on Using a Green-type formula for in we see thatIf we combine this with the limit relations (4.5), we get
A similar calculation for gives
It seems natural to do the following definitions.Definition 4.1. We define the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators in and asRemark 4.2. The inverses above exist at least on a space of finite codimension. This follows from the fact that is elliptic and is compact.Lemma 4.3. The operator is an elliptic pseudodifferential operator of order .
Proof. Using a resolvent identity, we see that It follows from the asymptotic expansion of in Lemma 2.1 and the fact that is a Schwartz kernel (again, see [18, Chapter 7, Section 11]) that is an elliptic pseudodifferential operator of order . Moreover the operator is compact, so the other two factors are pseudodifferential operators of order which do not change the principal symbol noticeably.
Let us now return to the expression of . We haveSince we are interested in and not , we may assume that and belong to . Then and . For such and we getor with the introduced operators aboveMoreover, is an elliptic pseudodifferential operator of order . This follows from the identity , and the fact that is an elliptic pseudodifferential operator of order . It follows from (4.13) that is an elliptic pseudodifferential operator or order .
Next, we prove the inequality (3.10). Because of the projections, it is enough to show it for functions in
The lower bound. We prove that there exists a subspace of finite codimension such that the lower bound in (3.10) is valid for all . Since we have so belongs to the kernel of the second-order elliptic operator . Let . We study the problemLet be the Schwartz kernel for . It is smooth away from the diagonal . One can repeat the theory with the single- and double-layer potentials for and write the solution in the case it exists.
Let be the double-layer operator evaluated at the boundary,The operator is compact, since the kernel has a weak singularity at the diagonal . Thus there exists a subspace of finite codimension such that the operator is invertible on . Hence, there exists a subspace of finite codimension where we have the representation formulafor all . The inequality follows easily from (4.16) for all such functions .
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 , so the third inequality is a standard estimate for elliptic operators.
5. Spectrum of Toeplitz Operators in A Reinhardt Domain
In the case when is a Reinhardt domain one can strengthen part of Lemma 3.6. Assume that , the interior of , is a Reinhardt domain. This means that and if , then the setis a subset of . If the setis convex in the usual sense, then is said to be logarithmically convex, and is a domain of holomorphy. Denote by the function defined byWe denote by the embedding operator. The -values , , of coincide with the numbers(we remind the reader of the notation of eigenvalues in Lemma 3.6). Unlike the case , see , it is natural to numerate the eigenvalues by the -tuples , just as for the eigenvalues of the Laplace operator in the unit cube , where the eigenvalues are given by .Lemma 5.1. Let and . Then
Proof. The denominator in (5.4) is easily calculated to be For the numerator, we do estimations from above and below, as in . First, note thatwhere is the transformed measure. It is clear thatFor the inequality in the other direction, fix . The hyperplanecuts in two components. Let be the component for which the inequality holds. Then we havewhere . It follows thatfrom which (5.5) follows.
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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