AMPAdvances in Mathematical Physics1687-91391687-9120Hindawi Publishing Corporation87370410.1155/2009/873704873704Research ArticleEigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann HamiltonianPerssonMikaelExnerPavelDepartment of Mathematical SciencesChalmers University of Technology and Göteborg University 412 96 GöteborgSwedenchalmers.se200910112008200927062008221020082009Copyright © 2009This 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.

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 , 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 . 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  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 2d. Let B>0 and denote by a the magnetic vector potentiala(x)=(a1(x),,a2d(x))=B2(x2,x1,x4,x3,,x2d,x2d1).It 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 byL=(ia)2and 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 operatorsLj=(i(x2j1,x2j)(a2j1,a2j))2,in the Hilbert spaces j=L2(2). Note that =j=1dj, andL=L1I(d1)+IL2I(d2)++I(d1)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:={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 eigenvaluesΛκ^=Bj=1d(2κj+1),κj.Note 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 . 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 derivativeszj=12(x2j1ix2j),z¯j=12(x2j1+ix2j).

We define creation and annihilation operators 𝒬j*, 𝒬j as𝒬j*=2ieWzjeW,𝒬j=2ieWz¯jeW,and note that[𝒬j*,𝒬k*]=[𝒬j,𝒬k]=[𝒬j*,𝒬k]=0,ifjk. 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

B2={ffisentireandd|f|2e(B/2)|z|2dm(z)<}.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 formu=|κ^|=μcκ^(𝒬*)κ^(eWfκ^),where (𝒬*)κ^=(𝒬1*)κ1(𝒬d*)κd and fκ^ all belong to B2. 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 decompositionsΛμ=|κ^|=μΛκ^,𝒫Λμ=|κ^|=μ𝒫Λκ^hold 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  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|xy|)+O(1),d=1,12π2|xy|2+O(log(1|xy|)),d=2,Γ(d1)2πd|xy|22d+O(|xy|42d),d>2,as |xy|0.

Proof.

The kernel Gρ(x,y) of Rρ can be written as Gρ(x,y)=0eρteLt(x,y)dm(t).Now, since the variables separate pairwise, we haveeLt(x,y)=j=1deLjt(x2j1,x2j,y2j1,y2j).The formula for eLjt is given in . It readseLjt=B4πexp(iB2(x2j1y2jx2jy2j1))1sinh(Bt/2)×exp(B4coth(Bt2)((x2j1y2j1)2+(x2jy2j)2)).Hence the formula for Gρ(x,y) becomesGρ(x,y)=(B4π)dexp(iB2j=1d(x2j1y2jx2jy2j1))I(|xy|2),whereI(s)=0eρt1sinhd(Bt/2)exp(B4coth(Bt2)s)dm(t).An expansion of I(s) shows thatI(s)~((2B)log(1s)+O(1),d=1,8B2s1+O(log(1s)),d=2,(4B)dΓ(d1)2s1d+O(s2d),d>2,ass0, from which (2.12) follows.

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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) byLΩ=(ia)2,inΩ, with magnetic Neumann boundary conditionsNu:=(ia)uν=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 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.

If d=1 then limj(j!(Λμlj(μ)))1/j=(B/2)(Cap(K))2.

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˜ asL˜=LKLΩ,inKΩ.The 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. LetR=R0=L1,R˜=L˜1=LK1LΩ1,and setV=R˜R,Tμ=𝒫ΛμV𝒫Λμ,forμ.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 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 byt1(μ)t2(μ),and the eigenvalues of R˜ in the interval (Λμ1,Λμ1+τ) byr1(μ)r2(μ).Lemma 3.4.

Given ε>0, there exists an integer l such that(1ε)tj+l(μ)rj(μ)Λμ1(1+ε)tjl(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 2d, 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 K0KK1 be compact domains such that KiΓ=. There exist a constant C>0 and a subspace 𝒮 of finite codimension such that1Cf,SμK0ff,TμfCf,SμK1ffor all f𝒮.

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+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 satisfylimj(j!tj(μ))1/j=B2(Cap(K))2if d=1, andn(λ,Tμ)~(μ+d1d1)1d!(|lnλ|ln|lnλ|)d,asλ0 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)Λμ1))1/j=B2(Cap(K))2if d=1, andN(Λμ1+λ,Λμ11,R˜)~(μ+d1d1)1d!(|lnλ|ln|lnλ|)d,asλ0.

If we translate this in terms of L˜ we getlimj(j!(Λμlj(μ)))1/j=B2(Cap(K))2for d=1, andN(Λμ1,Λμλ,L˜)~(μ+d1d1)1d!(|ln(λ/Λμ(Λμλ))|ln|ln(λ/Λμ(Λμλ))|)d~(μ+d1d1)1d!(|lnλ|ln|lnλ|),asλ0, 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  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 LL˜0, and hence V=R˜R0.

Next we prove the compactness of V. Let f and g belong to . 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=vKvΩ, and LKvKLΩvΩ=g. Integrating by parts and using (3.2) for vK and vΩ, we getf,Vg=f,R˜gRf,g=KLuvK¯dm(x)+ΩLuvΩ¯dm(x)KuLKvK¯dm(x)ΩuLΩvΩ¯dm(x)=ΓNu(vΩvK)¯dS.Here 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 u˜H2(2d) and v˜H2(2dΓ). However, the operator u˜Nu˜|Γ is compact as considered from H2(2d) to L2(Γ) and both v˜v˜Ω|Γ and v˜v˜K|Γ are compact as considered from H2(2dΓ) 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(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 that1CfL2(Γ)fH1(Γ)f,TμfCfL2(Γ)fH1(Γ)for all f in that subspace.

Let f and g belong to . Also, let u=Rf, v=R˜g, and w=Rg. We saw in (4.1) thatf,Vg=ΓNu(vΩvK)¯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(xy)α(y)dS(y),x2d,α(x)=ΓNyG0(xy)α(y)dS(y),x2dΓ,Aα(x)=ΓG0(xy)α(y)dS(y),xΓ,Bα(x)=ΓNyG0(xy)α(y)dS(y),xΓ.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 2d (see [18, Chapter 7, Section 11]), we have the following limit relations on Γ𝒜αK=AαK,αK=12α+Bα,𝒜αΩ=AαΩ,αΩ=12α+Bα.Using a Green-type formula for L in K we see thatβ=βK𝒜(NβK).If we combine this with the limit relations (4.5), we get(B12I)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=A1(B12I),(ND)K=(B12I)1A,(DN)Ω=A1(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(B12I)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 havef,Vg=ΓNu(vΩvK)¯dS=ΓNu(vΩw+wvK)¯dS=ΓNu((ND)Ω(N(vΩw)+(ND)K(N(wvK))))¯dS=ΓNu(((ND)K(ND)Ω)(Nw))¯dS.Since 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 getf,Vg=(Λμ)2ΓNf(((ND)K(ND)Ω)(Ng))¯dSor with the introduced operators abovef,Vg=(Λμ)2Γf((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 Λμ.

The lower bound. We prove that there exists a subspace 𝒮˜Λμ of finite codimension such that the lower bound in (3.10) is valid for all f𝒮˜. 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 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)=ΓNyE(x,y)α(y)dS(y),xΓ.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 𝒮˜Λμ of finite codimension where we have the representation formulaf(x)=ΓE(x,y)νy((12I+Bμ)1φ)(y)dS(y),xKfor all f𝒮˜. 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 inequalitiesfL2(Γ)fH1(Γ)CfH1/2(K)fH3/2(K)CfH2(K)2CfL2(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 0K and if zK, then the set{(w1,,wd)wj=tzj,t,|t|<1}is a subset of K. If the setlog|K|={(y1,,yd)yj=log|zj|,zK}is convex in the usual sense, then K is said to be logarithmically convex, and K is a domain of holomorphy. Denote by VK:d the function defined byVK(x)=supylog|K|x,y.We denote by J:B2˜:=L2(K,e(B/2)|z|2dm(z)) the embedding operator. The s -values sκ^, κ^d, of J coincide with the numbers{zκ^˜2zκ^B22}κ^0(we remind the reader of the notation of eigenvalues in Lemma 3.6). Unlike the case d=1, see , 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^B22=(2πB)d(2B)|κ^|κ^!.For the numerator, we do estimations from above and below, as in . First, note thatIκ^=zκ^˜2=log|K|exp(2κ^,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 hyperplaneκ^,x=(1δ)VK(κ^)cuts log|K| in two components. Let Pδ be the component for which the inequality κ^,x(1δ)VK(κ^) holds. Then we haveIκ^Pδexp(2|κ^|(1δ)VK(ω))dm˜(x)Cδexp(2|κ^|(1δ)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(1δ)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.

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