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

Size: px

Start display at page:

Download "Research Article Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian"

Eileen Lawrence
5 years ago
Views:

1 Hindawi Publishing Corporation Advances in Mathematical Physics Volume 009, Article ID , 5 pages doi:0.55/009/ Research Article Eigenvalue Asymptotics of the Even-Dimensional Exterior Landau-Neumann Hamiltonian Mikael Persson Department of Mathematical Sciences, Chalmers University of Technology and Göteborg University, 4 96 Göteborg, Sweden Correspondence should be addressed to Mikael Persson, mickep@chalmers.se Received 7 June 008; Accepted October 008 Recommended by Pavel Exner We study the Schrödinger operator with a constant magnetic field in the exterior of a compact domain in R d, d. 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. Copyright q 009 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.. 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,, that the spectrum of the Landau Hamiltonian consists of infinitely degenerate eigenvalues B q, q 0,,,..., 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.

2 Advances in Mathematical Physics 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. 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. 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, 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. and 3. 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, 0, to obtain the asymptotic formulas in Theorem 3.. In the higher dimensional case R d, d >, 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.. The Landau Hamiltonian in R d We denote by x x,...,x d apointinr d.letb>0 and denote by a the magnetic vector potential a x a x,...,a d x ) B x,x, x 4,x 3,..., x d,x d ).. It corresponds to an isotropic magnetic field of constant strength B. The Landau Hamiltonian L in R d describes a charged, spinless particle in this homogeneous magnetic field. It is given by L i a. and is essentially self-adjoint on the set C 0 Rd in the usual Hilbert space H L R d. For j,...,d,we also introduce the self-adjoint operators L j ) i x, ) ) a j x j j,a j,.3

3 Advances in Mathematical Physics 3 in the Hilbert spaces H j L R.NotethatH d j H j,and L L I d I L I d I d L d..4.. Landau Levels The spectrum of each two-dimensional Landau Hamiltonian L j consists of the so-called Landau levels, eigenvalues B q, q N : {0,,,...}, each of infinite multiplicity. Let κ κ,...κ d N d be a multi-index. We denote by κ κ κ d the length of the multiindex κ and also set κ! κ! κ d!. From.4 it follows that the spectrum of L consists of the infinitely degenerate eigenvalues d Λ κ B κj ), κ j N..5 j Note that Λ κ Λ κ if κ κ. Hence the spectrum of L consists of eigenvalues of the form Λ μ B μ d, μ N... 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 z,...,z d C d, where z j x j ix j. Also, we use the scalar potential W z B/4 z and the complex derivatives z j x i j x j ), i z j xj x j )..6 We define creation and annihilation operators Q j, Q j as Q j ie W z j ew, Q j ie W z j e W,.7 and note that [ Q j, ] [ ] [ Q k Qj, Q k Q j, Q k] 0, if j / k..8 The notation Q j for the creation operators is motivated by the fact that it is the formal adjoint of Q j in H.

4 4 Advances in Mathematical Physics A function u belongs to the lowest Landau level Λ 0 if and only if Q j u 0forj,...,d. This means that the function f e W u is an entire function, so via multiplication by e W the eigenspace L Λ0 corresponding to Λ 0 is equivalent to the Fock space } F B {f f is entire and f B/ z e dm z <..9 C d 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 form u c κ Q ) κ e W ) f κ, κ μ.0 where Q κ Q Q κ d κ d and f κ all belong to F B. The multiplicity of each eigenvalue Λ μ is infinite. We denote by P Λ κ and P Λμ the projection onto the eigenspaces L Λ κ and L Λμ, respectively, and note by.6 that the orthogonal decompositions L Λμ L Λ κ, P Λμ Λ κ. κ μ κ μp hold in H..3. The Resolvent Let R ρ L ρi 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. In Section 4., we will use the behavior of G ρ x, y near the diagonal x y, given in the following lemma. Lemma.. R ρ is an integral operator with kernel G ρ x, y that has the following singularity at the diagonal: π log G ρ x, y ) O, d, x y π x y O )) log, d, x y. d π d x y d O x y 4 d), d >, as x y 0. Proof. The kernel G ρ x, y of R ρ can be written as G ρ x, y 0 e ρt e Lt x, y dm t..3

5 Advances in Mathematical Physics 5 Now, since the variables separate pairwise, we have e Lt x, y d e L jt x j,x j,y j,y j). j.4 The formula for e L jt is given in 3. Itreads e Ljt B 4π exp ib x j y j x j y j )) sinh Bt/ exp B ) Bt x 4 coth j y j ) x j y j) ) )..5 Hence the formula for G ρ x, y becomes G ρ x, y ) B d exp ib 4π d x j y j x j y j )) I x y ),.6 j where I s 0 e ρt sinh d Bt/ exp B ) ) Bt 4 coth s dm t..7 An expansion of I s shows that ) ) log O, d, B s )) 8 I s B s O log, d, s ) 4 d d s d O s d), d >, B as s 0,.8 from which. follows. 3. The Exterior Landau-Neumann Hamiltonian in R d Let K R d be a simply connected compact domain with smooth boundary and let Ω R d \ K. We define the exterior Landau-Neumann Hamiltonian L Ω in H Ω L Ω by L Ω i a, in Ω, 3.

6 6 Advances in Mathematical Physics with magnetic Neumann boundary conditions N u : i a u ν 0, on. 3. Here ν denotes the exterior normal to. 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.. 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 μ l μ the eigenvalues of L Ω in the interval Λ μ, 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 4, Chapter. Theorem 3.. Let μ N. a If d then lim j j! Λ μ l μ j /j B/ Cap K. b If d>then N Λ μ, Λ μ λ, L Ω μ d ) d /d! ln λ / ln ln λ d as λ 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 actin different Hilbert spaces. We introduce the Hilbert space H K L K and define the interior Landau-Neumann Hamiltonian L K in H K by the same formulas as in 3. and 3. but with Ω replaced by K. WenotethatH H K H Ω and define L as L L K L Ω, in H K H Ω. 3.3 The inverse of L K is compact, so L K has at most a finite number of eigenvalues in each interval Λ μ, Λ μ. The operators L K and L Ω act in orthogonal subspaces of H, soσ L σ L K σ L Ω. This means that L has the same spectral asymptotics as L Ω in each interval Λ μ, Λ μ, so it is enough to prove the statements in Theorems 3. and 3. 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. Let R R 0 L, R L L K L Ω, 3.4 and set V R R, T μ P Λμ V P Λμ, for μ N. 3.5

7 Advances in Mathematical Physics 7 Lemma 3.3. V is nonnegative and compact. Proof. See Section 4.. By Weyl s theorem, the essential spectrum of R and R coincides. Since R R V and V 0, Theorem 3. follows immediately from 5, Theorem and the fact that σ R σ ess R {Λ μ }. We continue with the proof of Theorem 3.. Let τ>0besuch that Λ μ τ, Λ μ τ \{Λ μ } σ ess R. Denote the eigenvalues of T μ by t μ t μ, 3.6 and the eigenvalues of R in the interval Λ μ, Λ μ τ by r μ r μ. 3.7 Lemma 3.4. Given ε>0, there exists an integer l such that ε t μ j l r μ j Λ μ ε t μ, for all sufficiently large j. 3.8 j l Proof. See 4, Proposition.. 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 d, we define the Toeplitz operator S U μ as S U μ P Λμ χ U P Λμ, 3.9 where 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 K 0 K K be compact domains such that K i. There exist a constant C>0 and a subspace S Hof finite codimension such that f, S K 0 μ f f, T μ f C f, S K μ f C 3.0 for all f S. Proof. See Section 4.. The asymptotic expansion of the spectrum of S U μ is given in the following lemma.

8 8 Advances in Mathematical Physics Lemma 3.6. Denote by s μ s μ the eigenvalues of S U μ and by n λ, S U μ the number of eigenvalues of S U μ greater than λ counting multiplicity). Then a if d we have lim j j!s μ j /j B/ Cap U, b if d>we have n λ, S U μ μ d ) d /d! ln λ / ln ln λ d as λ 0. Proof. See 0, Lemma 3. for part a and 8, Proposition 7. for part b. Proof of Theorem 3.. We are now able to finish the proof of Theorem 3.. By letting K 0 and K in Lemma 3.5 get closer and closer to our compact K we see that the eigenvalues {t μ j } of T μ satisfy lim j j!t μ j )/j B Cap K ) 3. if d, and μ d n λ, T μ d ) d! ) ln λ d, as λ 0 3. ln ln λ if d>. Since neither formula 3. nor 3. is sensitive for finite shifts in the indices, it follows from Lemma 3.4 that the eigenvalues of {r μ j } R satisfy if d, and N Λ μ μ lim j! r j j λ, Λ μ, R ) Λ μ μ d d ))/j B ) Cap K 3.3 ) d! ) ln λ d, as λ ln ln λ If we translate this in terms of L we get lim j j! Λμ l μ j ))/j B Cap K ) 3.5 for d, and N Λ μ, Λ μ λ, L ) ) μ d ln λ/λμ Λ μ λ d d! ln ln λ/λ μ Λ μ λ ) ) μ d ln λ, as λ 0, d d! ln ln λ )d 3.6 for d>. This completes the proof of Theorem 3..

9 Advances in Mathematical Physics 9 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 6 and 7, Chapter 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. that L L 0, and hence V R R 0. Next we prove the compactness of V.Letf and g belong to H. Also, let u Rf and v Rg. Then u belongs to the domain of L and v belongs to the domain of L,sov v K v Ω, and L K v K L Ω v Ω g. Integrating by parts and using 3. for v K and v Ω,weget f, V g f, Rg Rf, g Lu v K dm x Lu v Ω dm x u L K v K dm x u L Ω v Ω dm x K Ω K Ω N u v ) Ω v K ds. 4. Here ds denotes the surface measure on. Take a smooth cutoff function χ C 0 Rd such that χ x 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.. By local elliptic regularity we have that ũ H R d and ṽ H R d \. However, the operator ũ N ũ is compact as considered from H R d to L and both ṽ ṽ Ω and ṽ ṽ K are compact as considered from H R d \ to L, so it follows that V is compact. 4.. Proof of Lemma 3.5 We start by showing that T μ, originally defined in L R d, can be reduced to an operator in L. More precisely we show that T μ can be realized as an elliptic pseudodifferential operator of order on some subspace of L of finite codimension, and hence there exists a constant C>0 such that C f L f H f, T μ f C f L f H 4. for all f in that subspace. Let f and g belong to H. Also, let u Rf, v Rg, and w Rg. Wesawin 4. that f, V g N u v ) Ω v K ds. 4.3

10 0 Advances in Mathematical Physics To go further we will introduce the Neumann-to-Dirichlet and Dirichlet-to-Neumann operators. Let G ρ x, y be as in.6. We start with the single- and double-layer integral operators, defined by Aα x G 0 x y α y ds y, x R d, Bα x Ny G 0 x y α y ds y, x R d \, Aα x G 0 x y α y ds y, x, Bα x Ny G 0 x y α y ds y, x. 4.4 The last two operators are compact on L, since,bylemma., their kernels have weak singularities. Moreover, since the kernel G 0 has the same singularity as the Green kernel for the Laplace operator in R d see 8, Chapter 7, Section, we have the following limit relations on Aα K Aα K, Bα K α Bα, Aα Ω Aα Ω, 4.5 Bα Ω α Bα. Using a Green-type formula for L in K we see that β Bβ K A N β K ). 4.6 If we combine this with the limit relations 4.5, weget B I ) β K A N β K ), on. 4.7 A similar calculation for Ω gives B I ) β Ω A N β Ω ), on. 4.8 It seems natural to do the following definitions.

11 Advances in Mathematical Physics Definition 4.. We define the Dirichlet-to-Neumann and Neumann-to-Dirichlet operators in K and Ω as DN K A B I ), ND K B I ) A, DN Ω A B I ), ND Ω B I ) A. 4.9 Remark 4.. 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. Proof. Using a resolvent identity, we see that ND K ND Ω B I ) B I ) A. 4.0 It follows from the asymptotic expansion of G 0 x, y in Lemma. and the fact that G 0 is a Schwartz kernel again, see 8, Chapter 7, Section that A is an elliptic pseudodifferential operator of order. 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 have f, V g N u v ) Ω v K ds N u v ) Ω w w v K ds N u ND Ω N vω w ) )))) ND K N w vk ds N u ) ND K ND Ω N w )) ds. 4. Since 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 f and w Rg Λ μ g. For such f and g we get f, V g Λ μ ) f ) N ND K ND Ω N g )) ds 4.

12 Advances in Mathematical Physics or with the introduced operators above f, V g Λ μ ) f DN ) K ND K ND Ω DN K g )) ds. 4.3 Moreover, DN K is an elliptic pseudodifferential operator of order. This follows from the identity A DN K B / I, and the fact that A is an elliptic pseudodifferential operator of order. It follows from 4.3 that T μ is an elliptic pseudodifferential operator or order. Next, we prove the inequality 3.0. 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 3.0 is valid for all f S. Since f L Λμ we have L μ f : L Λ μ f 0sof belongs to the kernel of the second-order elliptic operator L μ.letϕ f. We study the problem L μ f 0 in K, f ϕ on. 4.4 Let 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, B μ α x Ny E x, y α y ds y, x. 4.5 The operator B μ is compact, since the kernel Ny E x, y has a weak singularity at the diagonal x y. Thus there exists a subspace S L of finite codimension such that the operator / I B μ is invertible on S. Hence, there exists a subspace S L Λμ of finite codimension where we have the representation formula f x ) E x, y ν y I B μ ϕ) y ds y, x K 4.6 for all f S. The inequality f L K 0 C f L follows easily from 4.6 for all such functions f. Since we also have f L C f H the lower bound in 3.0 follows via the lower bound in 4.. The upper bound. By the upper bound in 4. it is enough to show the following inequalities f L f H C f H / K f H 3/ K C f H K C f L K. 4.7

13 Advances in Mathematical Physics 3 However, the first inequality is just the Trace theorem, the second is the Sobolev-Rellich embedding theorem. We note that L μ f 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 set { w,...,w d) w j tz j,t C, t < } 5. is a subset of K.Iftheset log K { y,...,y d) y j log z j, z K } 5. is 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 by V K x sup x, y. y log K 5.3 We denote by J : F B H : L K, e B/ z dm z the embedding operator. The s -values s κ, κ N d,ofjcoincide with the numbers { z κ H z κ F B } κ we remind the reader of the notation of eigenvalues in Lemma 3.6. Unlike the case d, see 0, it is natural to numerate the eigenvalues by the d-tuples κ κ,...,κ d, just as for the eigenvalues of the Laplace operator in the unit cube 0, d, where the eigenvalues are given by π d κ π d κ κ d. Lemma 5.. Let d> and ω κ/ κ. Then κ!s κ ) / κ B exp V K ω ) o ), as κ. 5.5 Proof. The denominator in 5.4 is easily calculated to be z κ F B π B ) d ) κ κ!. 5.6 B

14 4 Advances in Mathematical Physics For the numerator, we do estimations from above and below, as in. First,notethat I κ z κ H exp κ, x ) d m x, log K 5.7 where d m x is the transformed measure. It is clear that I κ exp κ V K ω ) m K. 5.8 For the inequality in the other direction, fix δ>0. The hyperplane κ, x δ V K κ 5.9 cuts log K in two components. Let P δ be the component for which the inequality κ, x δ V K κ holds. Then we have I κ P δ exp κ δ V K ω ) d m x C δ exp κ δ V K ω ), 5.0 where C δ P δ d m x > 0. It follows that ) ) / κ B d )/ κ B κ!s κ m K π exp V K ω ), ) ) / κ B d )/ κ B κ!s κ C δ π exp δ V K ω ), 5. 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. References V. Fock, Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld, Zeitschrift für Physik A, vol. 47, no. 5-6, pp , 98. L. Landau, Diamagnetismus der Metalle, Zeitschrift für Physik A, vol. 64, no. 9-0, pp , K. Hornberger and U. Smilansky, Magnetic edge states, Physics Reports, vol. 367, no. 4, pp , A. Pushnitski and G. Rozenblum, Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain, Documenta Mathematica, vol., pp , 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. 5, no. 3, pp , 990.

15 Advances in Mathematical Physics 5 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 , 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. 4, no. 0, pp , 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. 8, no. 3-4, pp , G. D. Raikov, Spectral asymptotics for the perturbed D 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 , N. Filonov and A. Pushnitski, Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains, Communications in Mathematical Physics, vol. 64, no. 3, pp , 006. G. Rozenblum and A. V. Sobolev, Discrete spectrum distribution of the landau operator perturbed by an expanding electric potential, American Mathematical Society Translations, Series, vol. 55, pp , 008. O. G. Parfënov, The singular values of the imbedding operators of some classes of analytic functions of several variables, Journal of Mathematical Sciences, vol. 7, no. 6, pp , B. Simon, Functional Integration and Quantum Physics, vol. 86 of Pure and Applied Mathematics, Academic Press, New York, NY, USA, N. S. Landkof, Foundations of Modern Potential Theory, vol. 80 of Die Grundlehren der mathematischen Wissenschaften, Springer, New York, NY, USA, M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Space, Mathematics and Its Applications, D. Reidel, Dordrecht, Holland, M. E. Taylor, Pseudodifferential Operators, vol.34ofprinceton Mathematical Series, Princeton University Press, Princeton, NJ, USA, M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich, Generalized Method of Eigenoscillations in Diffraction Theory, Wiley-VCH, Berlin, Germany, M. E. Taylor, Partial Differential Equations II: Qualitative Studies of Linear Equations, vol. 6 of Applied Mathematical Sciences, Springer, New York, NY, USA, 996.

The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria

ESI The Erwin Schrödinger International Boltzmanngasse 9 Institute for Mathematical Physics A-1090 Wien, Austria Remark on Magnetic Schrödinger Operators in Exterior Domains Ayman Kachmar Mikael Persson