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

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1 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 Vienna, Preprint ESI 186 (009) October, 009 Supported by the Austrian Federal Ministry of Education, Science and Culture Available online at

2 REMARK ON MAGNETIC SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS AYMAN KACHMAR AND MIKAEL PERSSON Abstract. We study the Schrödinger operator with a constant magnetic field in the exterior of a two-dimensional compact domain. Functions in the domain of the operator are subject to a boundary condition of the third type (Robin condition). In addition to the Landau levels, we obtain that the spectrum of this operator consists of clusters of eigenvalues around the Landau levels and that they do accumulate to the Landau levels from below. We give a precise asymptotic formula for the rate of accumulation of eigenvalues in these clusters, which appear to be independent from the boundary condition. 1. Introduction Magnetic Schrödinger operators in domains with boundaries appear in several areas of physics, one can mention the Ginzburg-Landau theory of superconductors, the theory of Bose-Einstein condensates, and of course the study of edge states in Quantum mechanics. We refer the reader to [1, 5, 10] for details and additional references on the subject. From the point of view of spectral theory, the presence of boundaries has an effect similar to that of perturbing the magnetic Schrödinger operator by an electric potential. In particular, in both cases, the essential spectrum consists of the Landau levels and the discrete spectrum form clusters of eigenvalues around the Landau levels. Several papers are devoted to the study of different aspects of these clusters of eigenvalues in domains with or without boundaries. In case of domains with boundaries, one can cite [6, 8, 9, 1, 11] for results in the semi-classical context, [17, 18] and the references therein for the study of accumulation of eigenvalues. Let us consider a compact and simply connected domain K R with a smooth C boundary. Let us denote by Ω = R \ K. Given a function γ L ( Ω) and a positive constant b (the intensity of the magnetic field), we define the Schrödinger operator L γ Ω,b with domain D(Lγ Ω,b ) as follows, D(L γ Ω,b ) = { u L (Ω) : ( iba 0 ) j u L (Ω), j = 1, ; ν Ω ( iba 0 )u + γu = 0 on Ω }, (1.1) 1991 Mathematics Subject Classification. 81Q10; 35PXX,47B5. Key words and phrases. Eigenvalue asymptotics, Landau levels, Boundary conditions, Magnetic field. 1

3 AYMAN KACHMAR AND MIKAEL PERSSON L γ Ω,b u = ( iba 0) u u D(L γ Ω,b ). (1.) Here, A 0 is the magnetic potential defined by A 0 (x 1, x ) = 1 ( x, x 1 ) (x 1, x ) R, (1.3) and ν Ω is the unit outward normal vector of the boundary Ω. The operator L γ Ω is actually the Freidrich s self-adjoint extension in L (Ω) associated with the semi-bounded quadratic form q γ Ω,B (u) = ( iba 0 )u dx + γ u ds, (1.4) Ω defined for all functions u in the form domain of q γ Ω, D(q γ Ω,b ) = H1 A 0 (Ω) = {u L (Ω) : ( iba 0 )u L (Ω)}. (1.5) Our result in the present paper is the following. Theorem 1.1. The essential spectrum of the operator L γ Ω,b consists of the Landau levels, σ ess (L γ Ω,b ) = {Λ n : n N}, Λ n = (n 1)b n N, (1.6) and for all ε (0, b) and n N, the spectrum of L γ Ω,b in the interval (Λ n, Λ n + ε) is finite. Moreover, for all n N, denoting by {l (n) j } j N the increasing sequence of eigenvalues of L γ Ω,b in the interval (Λ n 1, Λ n ), then lim j ( j!(λ n l (n) j ) Ω ) 1/j b( ) = Cap(K). (1.7) Here Cap(K) is the logarithmic capacity of the domain K = R \ Ω, and Λ 0 is set to be by convention. Roughly speaking, as in [18], the idea of the proof of Theorem 1.1 is to work with the resolvent of L γ Ω,b, which can be viewed as a perturbation of the resolvent of the Landau Hamiltonian in R. To get the asymptotic accumulation of the eigenvalues, we carry out a reduction to a boundary pseudo-differential operator, whose spectrum can be compared with that of Toeplitz operators. The paper is organized as follows. In Section we collect various auxiliary and technical results that will be useful in the proof. In Section 3, we give the proof of Theorem Preliminaries.1. Some abstract results. In this section we state three abstract results in operator theory that will be useful in the paper. Lemma.1. (Pushnitski-Rozenblum [18, Proposition.1]). Assume that A and B are two self-adjoint positive operators in L (R ) satisfying the following hypotheses:

4 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS 3 0 σ(a) σ(b). The form domain of A contains that of B, i.e. D(B 1/ ) D(A 1/ ). For all f D(B 1/ ), A 1/ f = B 1/ f, i.e. the quadratic forms of A and B agree on the form domain of B. Then, B A in the quadratic form sense, i.e. Bf, f Af, f f L (R ). Lemma.. Assume that A and B are two compact self-adjoint operators in L (R ), and S L (R ) a closed subspace of finite co-dimension, i.e. dim S <. If Af, f Bf, f f S, then there exists N 0 N such that λ n (A) λ n (B) n N 0. Here {λ n (T)} n N denotes the decreasing sequence of eigenvalues of a selfadjoint compact operator T. Proof. It suffices to prove that for a compact operator T, there exists a finite set F σ(t) such that σ(t S ) = σ(t) \ F. Here T S is the restriction of the operator T on the subspace S. Once this is shown to hold, the result of Lemma. follows from the variational min-max principle. Actually, writing T as the direct sum T S T S in L (R ) = S S, we get σ(t) = σ(t S ) σ(t S ). Here T S is the restriction of T on S which is actually a finite dimensional matrix. Hence σ(t S ) is finite and therefore, σ(t S ) = σ(t) \ σ(t S ) as required. The final abstract result we state is Theorem from [3]. Lemma.3. Assume A is a self-adjoint operator and V a compact and positive operator in L (R ) such that the spectrum of A in an interval (α, β) is discrete and does not accumulate at β, i.e. σ(a) (α, β) is finite. Then the spectrum of the operator B = A + V in (α, β) is discrete and does not accumulate at β... Some facts about the Landau Hamiltonian. In this section we review classical results concerning the Landau Hamiltonian L = ( iba 0 ) in R. (.1) Here A 0 is the magnetic potential with unit constant magnetic field introduced in (1.3), and B is a positive constant. The form domain of L is the magnetic Sobolev space H 1 A 0 (R ) = {u L (R ) : ( iba 0 )u L (R )}. The spectrum of L consists of infinitely degenerate eigenvalues called Landau levels, σ(l) = {Λ n : n N}, Λ n = (n 1)b n N.

5 4 AYMAN KACHMAR AND MIKAEL PERSSON We denote by L n the eigenspace associated with the Landau level Λ n, i.e. L n = ker(l Λ n ) n N. (.) We also denote by P n the orthogonal projection on the eigenspace L n. Notice that we can define R 0 = L 1, the resolvent of L. This is a bounded operator R 0 L(L (R )) with image in D(L). Furthermore, R 0 is an operator with integral kernel G 0 (x, y) [10]. We will need the following result from [17, Lemma.1] (see also [4]) on the behavior of G 0 near the diagonal. Lemma.4. R 0 is an integral operator with kernel G 0 (x, y) that has the following singularity at the diagonal, G 0 (x, y) 1 ( ) 1 π ln + O(1) as x y 0, (.3) x y and the corresponding behavior holds for G 0 (x, y)..3. Some boundary operators. Recall that K R has been assumed to be a compact simply connected subset of R and that we defined Ω = R \ K. Since Ω and K are complementary, the Hilbert space L (R ) is decomposed as the direct sum L (Ω) L (K) in the sense that any function u L (R ) can be represented as u Ω u K where u Ω and u K are the restrictions of u to Ω and K respectively. Denoting by the common boundary of Ω and K, we define the following operator on, u = N u + γ u = ν Ω ( ia 0 )u + γ u, (.4) where ν Ω is the unit outward normal vector to the boundary of Ω and A 0 is the magnetic potential from (1.3). The operator acts on functions in H 1 loc (Ω) or in H1 (K). We may write ( ) x in order to stress that the differentiation in (.4) is with respect to the variable x. Lemma.5. Define the following operators in L () Aα(x) = G 0 (x, y)α(y)ds(y) and Bα(x) = ( N ) y G 0 (x, y)α(y)ds(y), x. Here G 0 is the integral kernel from Lemma.4. Let u L (R ) be such that u Ω Hloc 1 (Ω) and u K H 1 (K). Then it holds that ( B + ( γ + 1 ) ) I u Ω = A( u Ω ) and ( ( B + γ 1 ) ) I u K = A( u K ).

6 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS 5 Proof. It is proved in [17, (4.6)-(4.7)] that, (B + 1 ) I u Ω = A( N u Ω ) and (B 1 ) I u K = A( N u K ). Inserting N = γi above, we get the formulae that we wish to prove. Remark.6. The operators A and B in Lemma.5 are well-defined bounded operators in L (). This is due to the behavior of the integral kernel G 0 from Lemma.4. Actually, for a fixed x R, the function G 0 (x ) L loc (R ). Moreover, since G 0 and G 0 are in L 1 loc (R R ), we see that the operators A and B from Lemma.5 are trace-class, hence compact in L (). Since A is a self-adjoint operator, we may define its regularized resolvent R A, { A 1 f if f (ker A) = Im A R A f = 0 if f ker A. For simplicity, we shall write A 1 for the regularized resolvent of A. Lemma.7. Assume γ L () satisfies 0 γ 0 < 1, where γ 0 = sup x γ(x). Then, there exists a closed subspace S in L (R ) such that, for any function u = u Ω u K S satisfying u Ω Hloc 1 (Ω) and u K H 1 (K), it holds that, u Ω = A 1 (B + u Ω = ( γ + 1 ) ) ( ( I u Ω, u K = A 1 B + γ 1 ) ) I u K, ( ( B + γ + 1 ) 1 ( ( I) A( u Ω ), u K = B + γ 1 ) 1 I) A( u K ). Here A and B are the operators from Lemma.5. Proof. The operators A and B, defined initially on L (), give rise to operators A ± and B ± defined in the Hilbert space H = H 1 A 0 (Ω) H 1 (K) via the trace operator on. Actually, for all u = u Ω + u K H, we define, A + u = Au Ω, A u = Au K, B + u = Bu Ω, B u = Bu K. We notice also that Sobolev embedding theorems give that A ± and B ± are compact. We shall simply write A and B for the operators A ± and B ± respectively. Notice that the operators (γ ± 1 )I are multiplication operators, hence (( σ γ ± 1 ) ) (( I = σ ess γ ± 1 ) ) (( I = Im γ ± 1 ) ) I. (( ) The hypothesis we made on γ guarantees that 0 σ ess γ ± 1 ) I. Since B is a compact operator, then Weyl s theorem gives σ ess (B + ( γ ± 1 ) I ) = σ ess (( γ ± 1 ) ) I.

7 6 AYMAN KACHMAR AND MIKAEL PERSSON ( ( ) Hence 0 σ ess B + γ ± 1 ) I. Therefore, taking (orthogonal is taken with respect to the scalar product of H), S = (S + + S ), S ± = [ ker ( ( B + γ ± 1 ) I)] H, we get that S has finite co-dimension in H, i.e. H = S V with dim H V <. We define S L (R ) to be the orthogonal of V with respect to the scalar product of L (R ), i.e. L (R ) = S V. We then get that S has finite co-dimension in L (R ). Now invoking Lemma.5 finishes the proof of the lemma. We conclude the section with the following lemma. Lemma.8. Let A be the operator from Lemma.5. Then A is an elliptic pseudo-differential operator of order 1. Proof. Since A is an operator with integral kernel G 0 (x, y) whose behavior near the diagonal is described in Lemma.4, the lemma follows from [7, Chapter 7, Section 11]..4. A result on Toeplitz operators. Recall the Landau levels {Λ n } n N together with their eigenspaces {L n } n N introduced in Section.. For all n N, we denoted by P n the orthogonal projector on the space L n. Given a positive integer n N and a compact simply connected domain U R with smooth boundary, the Toeplitz operator S U n is defined by, S U n = P n χ U P n in L (R ). (.5) Here χ U is the characteristic function of U. Since Im(χ U P n ) H 1 (U) and the boundary of U is smooth, then χ U P n is a compact operator, and so is the Toeplitz operator S U n. We state the following lemma which we take from [7, Lemma 3.]. Lemma.9. Given n N, denote by s (n) 1 s (n)... the decreasing sequence of eigenvalues of Sn U. Then, lim j ( (n)) 1/j b j!s j = (Cap(U)). 3. Proof of Theorem Proof under specific hypotheses. Recall the compact simply connected smooth domain K R and the exterior domain Ω = R \ K. We have introduced the operator L γ Ω,B with quadratic form qγ Ω,B from (1.4). We will use also the corresponding operator in K, namely L γ K,B. In this section we work under the following hypotheses: (H1) The quadratic forms q γ Ω,B and q γ K,B from (1.4) are strictly positive.

8 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS 7 (H) The function γ L () satisfies max x γ(x) < 1, (3.1) where is the common boundary of Ω and K. We treat the general case in Section 3.. When there is no ambiguity, we will skip B and γ from the notation, and write L Ω, L K, q Ω and q K for the operators L γ Ω,B, L γ K,B, the quadratic forms q γ Ω,B and q γ K,B respectively. Notice that, for all u = u Ω u K L (R ) such that u Ω D(q Ω ) and u K D(q K ), we have, q Ω (u Ω ) = ( iba 0 )u Ω dx + γ u Ω ds Ω q K (u K ) = ( iba 0 )u K dx γ u K ds. K If in addition, u H 1 A 0 (R ), then q Ω (u Ω )+q K (u K ) = R ( iba 0 )u dx. We point also that if u Ω D(L Ω ) and u K D(L K ), then u Ω = u K = 0, where is the trace operator from (.4) Extension of L Ω to an operator in L (R ). We pointed earlier that since Ω and K are complementary in R, the space L (R ) is decomposed as a direct sum L (Ω) L (K). This permits us to extend the operator L Ω in L (Ω) to an operator L in L (R ). Actually, let L = L Ω L K in D(L Ω ) D(L K ) L (R ). More precisely, L is the self-adjoint extension associated with the quadratic form q(u) = q Ω (u Ω ) + q K (u K ), u = u Ω u K L (R ). (3.) By our hypothesis (H1), we may speak of the resolvent R = L 1 of L. We then have the following lemma. Lemma 3.1. With L, R and L Ω defined as above, it holds that: (1) σ ess (L Ω ) = σ ess ( L). () λ σ ess ( R) \ {0} if and only if λ 0 and λ 1 σ ess (L Ω ). Proof. Since L = L Ω L K, then σ( L) = σ(l Ω ) σ(l K ). But K is compact and has a smooth boundary, hence it has a compact resolvent. Thus σ ess (K) = and here it follows the first assertion in the lemma above. Moreover, L Ω and L K are both strictly positive by hypothesis, hence 0 σ( L). It is then straight forward that σ ess ( L) = {λ R \ {0} : λ 1 σ ess ( R)}.

9 8 AYMAN KACHMAR AND MIKAEL PERSSON Essential spectrum of L Ω. With the operator L introduced above, we can view L Ω as a perturbation of the Landau Hamiltonian L in R introduced in (.1). Actually, we define V = R R 0 = L 1 L 1. Then we have the following result on the operator V. Lemma 3.. The operator V L(L (R )) is positive and compact. Moreover, for all f, g L (R ), it holds that f, V g = u (v Ω v K )ds, (3.3) where u = R 0 f and v = Rg. Proof. Notice that the form domain HA 1 0 (R ) of L is included in that of L, and that for u HA 1 0 (R ), we have q(u) = ( iba 0 )u dx. R Invoking Lemma.1, we get that the operator V is positive. Let us establish the identity in (3.3). Notice that f = Lu and g = Lv = L Ω v Ω L K v K. Then we have, f, V g = Lu v Ω dx + Lu v K dx u L Ω v Ω dx u L K v K dx. Ω K Ω K The identity in (3.3) then follows by integration by parts and by using the boundary conditions v Ω = v K = 0. Knowing that the trace operators are compact, we conclude from (3.3) that V is a compact operator. As corollary of Lemma 3., we get the first part of Theorem 1.1 proved. Corollary 3.3. Assume the hypothesis (H1) above holds. Then and for all ε (0, b), σ ess (L Ω ) = {λ n : n N} Λ n = (n 1)b, σ(l Ω ) (Λ n, Λ n + ε) is finite n N. Proof. Invoking Lemma 3.1, it suffices to prove that σ ess ( R) = {Λ 1 n : n N} in order to get the result concerning the essential spectrum of L Ω. Notice that R = R 0 + V with V a compact operator. Hence by Weyl s theorem, σ ess ( R) = σ ess (R 0 ) = σ ess (L 1 ). But we know from Section. that σ ess (R 0 ) = {Λ 1 n : n N} as was required to prove. Since the operator V is compact and positive, invoking Lemma.3, we get that σ( R) (Λ 1 n ε, Λ 1 n ) is finite. Since R = L 1, this gives that σ(l Ω ) (Λ n, Λ n + ε) is finite.

10 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS Reduction to a Toeplitz operator. In light of Corollary 3.3, we have only to establish the second part of Theorem 1.1, namely the asymptotic formulae in (1.7). Let n N and pick τ > 0 such that ( (Λ 1 n τ, Λ 1 n + τ) \ {Λ 1 n } ) σ ess ( R) =. Denote by {r (n) j } j 1 the decreasing sequence of eigenvalues of R in the interval (Λ 1 n, Λ 1 n + τ). In order to prove (1.7), it suffices to show that ( ( lim j! r (n) j j We introduce the operator )) 1/j Λ 1 b n = (Cap(K)). (3.4) T n = P n V P n, (3.5) where P n is the orthogonal projection on the eigenspace L n associated with Λ n. By Lemma 3., V is a compact operator, hence T n is also a compact operator. Denote by {t (n) j } the decreasing sequence of eigenvalues of T n. The next lemma, proved in [18, Proposition.], shows that r (n) j Λ 1 n are close to the eigenvalues of T n. Lemma 3.4. Given ε > 0 there exist integers l and j 0 such that (1 ε)t (n) j+l r(n) j Λ 1 n (1 + ε)t (n) j l, j j 0. The spectrum of T n will be further related to the spectra of Toeplitz operators. Recall that associated with a compact domain U R, we introduced in (.5) the Toeplitz operator S U n. We will prove the following result. Lemma 3.5. Let K 0 K K 1 be compact domains with K i K =. There exist a constant C > 0 and a subspace S L (R ) of finite codimension such that 1 C f, SK 0 n f f, T n f C f, S K 1 n f f S. (3.6) The proof of Lemma 3.5 is by reduction of the operator T n to a pseudodifferential operator on the common boundary of Ω and K. We will give the proof in the next section, but we give first the proof of (3.4). Corollary 3.6. Assume the hypotheses (H1) and (H) above hold. Then the claim in (3.4) above is true. Proof. Invoking Lemma., the result of Lemma 3.5 provides us with a sufficiently large integer j 0 N such that, for all j j 0, we have, 1 C s(n) j,k 0 t (n) j Cs (n) j,k 1. Here {s (n) j,k 0 } j and {s (n) j,k 1 } j are the decreasing sequences of S K 0 n and S K 1 n respectively. Implementing the result of Lemma.9 in the inequality above,

11 10 AYMAN KACHMAR AND MIKAEL PERSSON we get b ( ) (Cap(K 0)) lim j!t (n) 1/j b j j (Cap(K 1)). Since both K 0 K K 1 are arbitrary, we get by making them close to K, ( ) lim j!t (n) 1/j b j j = (Cap(K)). Implementing the above asymptotic limit in the estimate of Lemma 3.4, we get the announced result in Corollary 3.6 above. Summing up the results of Corollaries 3.3 and 3.6, we end up with the proof of Theorem 1.1, provided the hypotheses (H1) and (H) are verified. Corollary 3.7. Assume that hypotheses (H1) and (H) above are satisfied. Then the results of Theorem 1.1 hold true Reduction to a boundary pseudo-differential operator. This section is devoted to the proof of Lemma 3.5. We start with the following reduction of the operator T n from (3.5). Lemma 3.8. There exists a subspace S L (R ) with finite co-dimension such that, for all f, g S, it holds that, f, T n g = 1 Λ (P n f) T((P n g)) ds. (3.7) n Here, T = T B, A 1 T 1 B,+ ( (, T B,± = B + γ ± 1 ) ) I and A, B the operators from Lemma.5. Proof. Recall that R 0 = L 1 is the resolvent of the Landau Hamiltonian, R 0 that of the Hamiltonian L = L Ω L K. We denote by u = R 0 P n f = Λ 1 n P n f, v = RP n g = v Ω v K and w = R 0 P n g = Λ 1 n P n g. Notice that f, T n g = P n f, V P n g where V is the operator from (3.3). Invoking Lemma 3., we write, P n f, V P n g = u (v Ω v K )ds = u (v Ω w + w v K )ds But Lemma.7 provides us with a subspace S L (R ) with finite codimension so that we can write, P n f, V P n g = u (T 1 B, A( (v Ω w)) + T 1 B,+ A( (w v K))) ds.,

12 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS 11 Notice that v Ω and v K are in the domain of the operators L Ω and L K respectively, hence v Ω = v K = 0. Consequently, P n f, V P n g = u (T 1 B,+ T 1 B,+ )A( w))ds. (3.8) Lemma.7 also gives, w = A 1 T B, w, u = A 1 T B, u. Inserting this in (3.8), we get by a simple calculation, P n f, V P n g = u Tw ds, where T is the operator introduced in Lemma 3.8 above. Recalling that u = R 0 P n f = Λ 1 n P n f and w = R 0 P n g = Λ 1 n P n g, we get the identity announced in Lemma 3.8 above. Lemma 3.9. There exist a constant C > 1 and a subspace S L (R ) such that, for all f S it holds that, 1 C f L () f H 1 () f, T n f C f L () f H 1 (). Proof. Lemma.8 says that A is an elliptic pseudo-differential operator of order 1, hence A 1 is a pseudo-differential operator of order 1. On the other hand, since B is compact with integral kernel, we know that B is a pseudo-differential operator of order 0, and the same holds true for the projector P n on the Landau level (it is also an operator with smooth integral kernel). Therefore, the operator T = P n TP n with T the operator from Lemma 3.8 is a pseudo-differential operator of order 1. From Lemma 3.8 we get the existence of the subspace S and T n restricted to S L () becomes a pseudo-differential operator of order 1. Thus, there exists a positive constant C such that the double inequality announced in the above lemma holds for all f S. Proof of Lemma 3.5. Step 1. Lower bound. We prove that there exists a subspace S L (R ) with finite co-dimension such that the lower bound in (3.6) is valid for all f S. We take the subspace S to be that given in Lemma 3.9. Let u = P n f, with P n the orthogonal projection on the eigenspace L n associated with the Landau level Λ n. Since L n is infinite dimensional, then there exists a finite dimensional space V such that L n \ V S. So by possibly changing S to a smaller subspace, we may assume that P n S S and hence u S. By the definition of T n from (3.5), the estimate of Lemma 3.9 gives, So it suffices to prove that f, T n f 1 C u L () u H 1 (). f, S K 0 n C u L () u H 1 (),

13 1 AYMAN KACHMAR AND MIKAEL PERSSON for some positive constant C. Recalling the definition of S K 0 n, this is equivalent to showing that u L (K 0 ) C u L () u H 1 (). (3.9) Notice that L n u = 0, where L n = L Λ n I. Let us denote by E(x, y) the Green s potential of the operator L n. Then E is smooth away from the diagonal x = y and decays near the diagonal in the same way described in Lemma.4 (see Stampacchia [5]). Let B n be the double layer operator evaluated at the boundary, i.e. for any α Hloc 1 (R ), B n α(x) = ( N ) y E(x, y)α(y)ds(y), x. Here we remind the reader that N = ν Ω ( iba 0 ) and ν Ω is unit outward normal of the boundary Ω =. The operator B n is compact, since the kernel E(x, y) has a logarithmic singularity at the diagonal x = y. Therefore we can invert the operator B n +(γ + 1 )I in a subspace S L (R ) with finite co-dimension (thanks in particular to our hypothesis (H) on γ). That way, similar to Lemma.5, using the results in [7, Chapter 7, Section 11], we can write, u(x) = ( N ) y E(x, y) ( B n + ( γ + 1 ) I) 1 u(y)ds(y), x K, (3.10) for all u P n S. Changing possibly the space S from Lemma 3.9, we may assume that S is contained in the new space S. In this case, for all u S, (3.10) gives us the inequality u L (K 0 ) C u L (). This is now sufficient deduce the estimate in (3.9) above. Step. Upper bound. Now we establish the upper bound in (3.6). This part is actually quiet easy. Let S be the same subspace obtained for the derivation of the lower bound in the previous paragraph. Let f S and u = P n f, the projection on the eigenspace L n. Notice that the trace theorem gives, u L () u H 1 () C u H 1/ (K) u H 3/ (K), for some positive constant C. Invoking the Sobolev-Rellich embedding theorem, we get for a possibly new constant C, u L () u H 1 () C u H (K). Notice that L n u = Lu Λ n u = 0. Then by elliptic regularity, given a domain K 1 such that K K 1, there exists a constant C K1 such that, u H () C K1 ( Ln u L (K 1 ) + u L (K 1 )) = CK1 u L (K 1 ). Summing up, we get, P n f L () P n f H 1 () C P n f L (K 1 ), f S.

14 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS 13 Substituting the above inequality in the estimate of Lemma 3.9, we get the upper bound announced in (3.6). 3.. Proof of Theorem 1.1: General case. Notice that we have already proved Theorem 1.1 under the extra hypotheses (H1)-(H). Since the quadratic form from (1.4) is semi-bounded, we can eliminate the Hypothesis (H1) on positivity of the quadratic forms. The price to pay is a simple shift by a positive constant of the operator, which does not affect the asymptotic behavior of eigenvalues. So we only have to show how we can eliminate the hypothesis (H) on the function. For each l > 0, we denote by Ω l = {lx : x Ω}. Making the change of scale y = lx, x Ω, we get the following lemma. Lemma For every l > 0, let b l = B/(l ) and γ l (x) = (γ(x/l))/l for all x Ω l. Then the spectrum of the operator L γ l Ω l,b l is given by the spectrum of the operator L γ Ω,b as follows: ( ) σ L γ l Ω l,b l = 1 ) (L l σ γ Ω,b. Let γ 0 = max x Ω γ(x). We may choose l sufficiently large such that γ 0 /l < 1. In this case, the operator Lγ l Ω l,b l satisfies the hypothesis (H). Hence the result of Theorem 1.1 holds true for the operator L γ l Ω l,b l. Invoking Lemma 3.10, we deduce that the results in Theorem 1.1 hold true for the operator L γ Ω,b (we actually use that Cap(Ω l) = lcap(ω)). Acknowledgements AK is supported by a Starting Independent Researcher grant by the ERC under the FP7. MP is supported by the Lundbeck Foundation. References [1] A, Aftalion, B. Helffer. On mathematical models for Bose-Einstein condensates in optical lattices. Rev. Math. Phys. 1 () 9-78 (009). [] M. S. Agranovich, B. Z. Katsenelenbaum, A. N. Sivov, and N. N. Voitovich. Generalized method of eigenoscillations in diffraction theory. WILEY-VCH Verlag Berlin GmbH, Berlin, Translated from the Russian manuscript by Vladimir Nazaikinskii. [3] M. Sh. Birman and M. Z. Solomjak. Spectral theory of selfadjoint operators in Hilbert space. Mathematics and its Applications (Soviet Series). D. Reidel Publishing Co., Dordrecht, Translated from the 1980 Russian original by S. Khrushchëv and V. Peller. [4] V. Fock. Bemerkung zur Quantelung des harmonischen Oszillators im Magnetfeld. Z. Phys, 47: , 198. [5] S. Fournais, B. Helffer. Spectral Methods in Surface Superconductivity. Monograph to appear (009). [6] S. Fournais, A. Kachmar. On the energy of bound states for magnetic Schrödinger operators. J. Lond. Math. Soc. 4 (009). DOI:10.111/jlms/jdp08 [7] N. Filonov and A. Pushnitski. Spectral asymptotics of Pauli operators and orthogonal polynomials in complex domains. Comm. Math. Phys., 64(3):759 77, 006.

15 14 AYMAN KACHMAR AND MIKAEL PERSSON [8] R. Frank. On the asymptotic number of edge states for magnetic Schrödinger operators. Proc. London Math. Soc. (3) 95 (1) 1-19 (007). [9] B. Helffer, A. Morame. Magnetic bottles in conncetion with superconducitivity. J. Func. Anal. 181 () (001). [10] K. Hornberger, U. Smilansky. Magnetic edge states. Phys. Rep., 367(4):49 385, 00. [11] A. Kachmar. Weyl asymptotics for magnetic Schrödinger operators and de Gennes boundary condition. Rev. Math. Phys. 0 No. 8 (008), [1] A. Kachmar. On the ground state energy for a magnetic Schrödinger operator and the effect of the De Gennes boundary condition. J. Math. Phys. 47 (7) (3 pp.) 006. [13] L. Landau. Diamagnetismus der Metalle. Z. Phys, 64:69 637, [14] N. S. Landkof. Foundations of modern potential theory. Springer-Verlag, New York, 197. Translated from the Russian by A. P. Doohovskoy, Die Grundlehren der mathematischen Wissenschaften, Band 180. [15] M. Melgaard and G. Rozenblum. Eigenvalue asymptotics for weakly perturbed Dirac and Schrödinger operators with constant magnetic fields of full rank. Comm. Partial Differential Equations, 8(3-4): , 003. [16] O. G. Parfënov. The singular values of the imbedding operators of some classes of analytic functions of several variables. J. Math. Sci., 7(6): , Nonlinear boundary-value problems. Differential and pseudo-differential operators. [17] M. Persson. Eigenvalue asymptotics for the even-dimensional exterior Landau- Neumann Hamiltonian. Advances in Mathematical Physics. Volume 009 (009). Article ID pages. [18] A. Pushnitski, G. Rozenblum. Eigenvalue clusters of the Landau Hamiltonian in the exterior of a compact domain. Doc. Math., 1: , 007. [19] G.D. Raĭkov. Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips. Comm. Partial Differential Equations, 15(3): , [0] 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 Process. Related Fields, 9(4): , 003. [1] G. Rozenblum, A.V. Sobolev. Discrete spectrum distribution of the landau operator perturbed by an expanding electric potential. To appear in Contemporary Mathematics, AMS, 008. [] G.D. Raikov, S. Warzel. Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials. Rev. Math. Phys., 14(10): , 00. [3] G.D. Raikov, S. Warzel. Spectral asymptotics for magnetic Schrödinger operators with rapidly decreasing electric potentials. C. R. Math. Acad. Sci. Paris, 335(8): , 00. [4] B. Simon. Functional integration and quantum physics, volume 86 of Pure and Applied Mathematics. Academic Press Inc. [Harcourt Brace Jovanovich Publishers], New York, [5] G. Stampacchia. Équations elliptiques du second ordre à coefficients discontinus. Séminaire de Mathématiques Supérieures, No. 16 (Été, 1965). Les Presses de l Universit de Montral, Montreal, Que. 36 pp. 1966). [6] M.E. Taylor. Pseudodifferential operators, volume 34 of Princeton Mathematical Series. Princeton University Press, Princeton, N.J., [7] M.E. Taylor. Partial differential equations. II, volume 116 of Applied Mathematical Sciences. Springer-Verlag, New York, Qualitative studies of linear equations.

16 SCHRÖDINGER OPERATORS IN EXTERIOR DOMAINS 15 (A. Kachmar and M. Persson) Aarhus University, Department of Mathematical Sciences, 1530 Ny Munkegade, 8000 Aarhus C, Denmark address, A. Kachmar: address, M. Persson: