Clusters of eigenvalues for the magnetic Laplacian with Robin condition
aa r X i v : . [ m a t h . SP ] J un CLUSTERS OF EIGENVALUES FOR THE MAGNETICLAPLACIAN WITH ROBIN CONDITION
MAGNUS GOFFENG, AYMAN KACHMAR, AND MIKAEL PERSSON SUNDQVIST
Abstract.
We study the Schrödinger operator with a constant magnetic fieldin the exterior of a compact domain in euclidean space. Functions in thedomain of the operator are subject to a boundary condition of the third type(a magnetic Robin condition). In addition to the Landau levels, we obtainthat the spectrum of this operator consists of clusters of eigenvalues aroundthe Landau levels and that they do accumulate to the Landau levels frombelow. We give a precise asymptotic formula for the rate of accumulation ofeigenvalues in these clusters, which is independent of the boundary condition. Introduction
Magnetic Schrödinger operators in domains with boundaries appear in severalareas of physics. E.g. the Ginzburg–Landau theory of superconductors, the theoryof Bose–Einstein condensates, and of course the study of edge states in Quantummechanics. We refer the reader to [1, 10, 16] for details and additional references onthe subject. From the point of view of spectral theory, the presence of boundarieshas an effect similar to that of perturbing the magnetic Schrödinger operator byan electric potential. In both cases, the essential spectrum consists of the Landaulevels and the discrete spectrum form clusters of eigenvalues around the Landaulevels. Several papers are devoted to the study of different aspects of these clustersof eigenvalues in domains with or without boundaries. For results in the semi-classical context, see [11, 12, 14, 17, 18]. In case of domains with boundaries,see [21, 22].Let us consider a compact domain K ⊂ R d with Lipschitz boundary. Let usdenote by K ◦ the interior of K , Ω = R d \ K and ∂ Ω the common boundary ofΩ and K . Given a real valued function τ ∈ L ∞ ( ∂ Ω , R ) and a positive constant b (the intensity of the magnetic field), we define the Schrödinger operator L τ Ω ,b withdomain D ( L τ Ω ,b ) as follows, D (cid:0) L τ Ω ,b (cid:1) = (cid:8) u ∈ L (Ω) : ( ∇ − ib A ) j u ∈ L (Ω) , j = 1 , ν Ω · ( ∇ − ib A ) u + τ u = 0 on ∂ Ω (cid:9) , (1.1) L τ Ω ,b u = − ( ∇ − ib A ) u ∀ u ∈ D (cid:0) L τ Ω ,b (cid:1) . (1.2)Here, A is the magnetic potential in the symmetric gauge defined by A ( x , x , . . . , x d ) = ( − x , x , . . . , − x d , x d − ) , (1.3)and ν Ω is the unit outward normal vector of the boundary ∂ Ω. We also introducethe boundary Neumann and Robin differential notations ∂ N = ν Ω · ( ∇ − ib A ) , and ∂ R = ∂ N + τ = ν Ω · ( ∇ − ib A ) + τ. (1.4) Mathematics Subject Classification.
Key words and phrases.
Eigenvalue asymptotics, Landau levels, Boundary conditions, Mag-netic field.
The operator L τ Ω ,b is actually the Friedrich’s self-adjoint extension in L (Ω) associ-ated with the semi-bounded quadratic form l τ Ω ,b ( u ) = Z Ω | ( ∇ − ib A ) u | dx + Z ∂ Ω τ | u | dS , (1.5)defined for all functions u in the form domain D (cid:0) l τ Ω ,b (cid:1) = H A (Ω) = (cid:8) u ∈ L (Ω) : ( ∇ − ib A ) u ∈ L (Ω) (cid:9) ⊆ H (Ω) . (1.6)We introduce the (multidimensional) Landau levels Λ q , q ∈ N , asΛ q := (cid:0) q −
1) + d (cid:1) b, q ∈ N \ { } , Λ := −∞ . The name is motivated by the fact that these numbers (for q ∈ N \ { } ) are theeigenvalues of the Landau Hamiltonian in R d , see Section 2.2.We are now able to state the first main result, concerning the essential spectrumof L τ Ω ,b together with the non-accumulation of eigenvalues to the Landau levels fromabove. Theorem 1.1.
Let Ω ⊂ R d be a compactly complemented Lipschitz domain and τ ∈ L ∞ ( ∂ Ω , R ) . The essential spectrum of the operator L τ Ω ,b consists of the Landaulevels, σ ess (cid:0) L τ Ω ,b (cid:1) = (cid:8) Λ q : q ∈ N \ { } (cid:9) . (1.7) Moreover, for all ε ∈ (0 , b ) and q ∈ N \{ } , the spectrum of L τ Ω ,b in the open interval (cid:0) Λ q , Λ q + ε (cid:1) is finite. Next, we restrict our attention to the case that ∂ Ω is C ∞ and τ ∈ C ∞ ( ∂ Ω , R ).For ε >
0, we let N (cid:0) Λ q − , Λ q − ε, L τ Ω ,b (cid:1) denote the number of eigenvalues of L τ Ω ,b inthe open interval (cid:0) Λ q − , Λ q − ε (cid:1) , counting multiplicity. We also denote by Cap( K )the logarithmic capacity of the domain K = R \ Ω. Theorem 1.2.
Let Ω ⊂ R d be a compactly complemented domain with smoothboundary and τ ∈ C ∞ ( ∂ Ω , R ) . (A) Assume d = 1 . For all q ∈ N \ { } , denoting by (cid:8) ℓ ( q ) j (cid:9) j ∈ N the nondecreas-ing sequence of eigenvalues of L τ Ω ,b (counting multiplicities) in the interval (Λ q − , Λ q ) , the following holds: lim j → + ∞ (cid:16) j ! (cid:0) Λ q − ℓ ( q ) j (cid:1)(cid:17) /j = b (cid:0) Cap( K ) (cid:1) . (1.8)(B) Assume d ≥ . The counting function N (cid:0) Λ q − , Λ q − ε, L τ Ω ,b (cid:1) has the asymp-totics: N (cid:0) Λ q − , Λ q − ε, L τ Ω ,b (cid:1) ∼ (cid:18) q + d − d − (cid:19) d ! (cid:18) | log ε | log | log ε | (cid:19) d as ε → + . (1.9) Remark 1.3.
Theorems 1.1 and 1.2 were obtained for the Neumann case ( τ ≡
0) bythe third author in [21], and our proofs will follow the same idea as in [21]. However,in [21], the full details concerning the reduction to Toeplitz type operators were notwritten out explicitly. In this paper, we aim not only to generalize the Neumannresult, but also to make the proof of Theorem 1.2 more transparent.
Remark 1.4.
Our proof of Theorem 1.2 is carried out for τ being a self-adjointpseudo-differential operator of order 0 on ∂ Ω. They can be generalized to self-adjoint pseudo-differential operators of order t <
1. We also note that the pseudo-differential nature of the proof of Theorem 1.2 requires a fair amount of regularityon the boundary. Any considerable reduction of the regularity assumptions inTheorem 1.2 would require a new approach or a perturbation result for the lefthand side of Equation (1.8) and (1.9).
LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 3
The rest of the paper is devoted to the proof of Theorem 1.1 and Theorem 1.2.The proof of Theorem 1.1 is contained in Section 2. The proof of Theorem 1.2 isdivided into two sections: the bulk of the proof is contained in Section 3 except for atechnical lemma, Lemma 3.14, which is proved in Section 4. Similarly to [21, 22], themain idea in both proofs is to compare the resolvent of L τ Ω ,b with the resolvent of theLandau Hamiltonian. Roughly speaking, the resolvents are compact perturbationof one another.Section 2 goes analogously to [21, Section 3 . H / -norm on the boundary. It isproven via standard pseudo-differential techniques in Section 4.2. Proof of Theorem 1.1
In this section we prove Theorem 1.1. As remarked above, the proof goes alongthe lines of [21, Section 3.1]. After adding on a Landau Hamiltonian L − τK,b on K (the sign of − τ comes from the orientation on the boundary) we can consider anoperator densely defined in L ( R d ) coinciding in form sense on the form domainof the usual Landau Hamiltonian. While L − τK,b has discrete spectrum, the proof ofTheorem 1.1 is deduced below in Corollary 2.6 from the abstract results from [6]and [22] reviewed in the next subsection.2.1. Two abstract results.
In this section we state two abstract results. We willuse the first result to conclude positivity of difference of resolvents and the secondone to obtain the finiteness of eigenvalues above each Landau level.
Lemma 2.1 (Pushnitski-Rozenblum [22, Proposition 2.1]) . Assume that A and B are two self-adjoint positive operators satisfying the following hypotheses: • σ ( A ) ∪ σ ( B ) . • The form domain of A contains that of B , i.e. D ( B / ) ⊂ D ( A / ) . • For all f ∈ D ( B / ) , (cid:13)(cid:13) A / f (cid:13)(cid:13) = (cid:13)(cid:13) B / f (cid:13)(cid:13) , 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. (cid:10) B − f, f (cid:11) ≤ (cid:10) A − f, f (cid:11) ∀ f. Lemma 2.2 ([6, Theorem 9.4.7]) . Assume A is a self-adjoint operator and V acompact and positive operator such that the spectrum of A in an interval ( α, β ) isdiscrete and does not accumulate at β . 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 R d . In this section wereview classical results concerning the Landau Hamiltonian L = − ( ∇ − ib A ) in R d . (2.1) MAGNUS GOFFENG, AYMAN KACHMAR, AND MIKAEL PERSSON SUNDQVIST
Here A is the magnetic potential of a unit constant magnetic field of full rankintroduced in (1.3), and b is a positive constant. The form domain of L is themagnetic Sobolev space H A ( R d ) = (cid:8) u ∈ L ( R d ) : ( ∇ − ib A ) u ∈ L ( R d ) (cid:9) ⊆ H ( R d ) . The spectrum of L consists of infinitely degenerate eigenvalues called Landau levels, σ ( L ) = σ ess ( L ) = (cid:8) Λ q : q ∈ N \ { } (cid:9) . We denote by L q the eigenspace associated with the Landau level Λ q , i.e. L q = Ker( L − Λ q ) ∀ q ∈ N \ { } . (2.2)We use the notation P q for the orthogonal projection onto the eigenspace L q .The operator L can be expressed in terms of creation and annihilation operators.We introduce the complex notation z j = x j − + ix j , j = 1 , . . . , d and let Ψ = b | z | be a scalar potential for the magnetic field, i.e. ∆Ψ = b . The differential expressions Q j = − ie − Ψ ∂∂z j e Ψ , Q j = − ie Ψ ∂∂z j e − Ψ formally satisfy the following well known identities: Q j = Q ∗ j , ≤ j ≤ d, (cid:2) Q j , Q k (cid:3) = 2 bδ jk , ≤ j, k ≤ d,L = d X j =1 Q j Q j + bd. (2.3)2.3. Extension of L Ω to an operator in L ( R d ) . We introduced the operator L τ Ω ,b with quadratic form l τ Ω ,b in (1.5). We will use also the corresponding operatorin K ◦ , namely L − τK,b . We will throughout the paper work under the assumptionthat the two quadratic forms l τ Ω ,b and l − τK,b are strictly positive. This is alwaysattainable after a shift of the quadratic forms by a constant. The effect of theconstant is merely a shifting of the spectrums of all involved operators, hence wewill for notational simplicity always assume that this constant is 0. Remark 2.3.
Notice that, for u ∈ H A ( R d ), l τ Ω ,b ( u Ω ) + l − τK,b ( u K ) = Z R d | ( ∇ − ib A ) u | dx. This motivates the usage of − τ for the quadratic form on K . For u Ω ∈ D ( L τ Ω ,b )and u K ∈ D ( L − τK,b ), ∂ R u Ω = ∂ R u K = 0 on ∂ Ω , where ∂ R denotes the Robin differential expression from (1.4).When there is no ambiguity, we will skip b and τ from the notation, and write L Ω , L K , l Ω and l K for the operators L τ Ω ,b , L − τK,b , the quadratic forms l τ Ω ,b and l − τK,b respectively.Since Ω and K are complementary in R d , the space L ( R d ) is decomposed asa direct sum L (Ω) ⊕ L ( K ). This permits us to extend the operator L Ω in L (Ω)to an operator e L in L ( R d ). We let e L = L Ω ⊕ L K in D ( L Ω ) ⊕ D ( L K ) ⊂ L ( R d ).More precisely, e L is the self-adjoint extension associated with the quadratic form e l ( u ) = l Ω ( u Ω ) + l K ( u K ) , u = u Ω ⊕ u K ∈ L ( R d ) , u Ω ∈ D ( l Ω ) , u K ∈ D ( l K ) . (2.4)Since l Ω and l K are strictly positive, we may speak of the inverse e L − of e L . Wehave the following lemma. LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 5
Lemma 2.4.
With e L and L Ω defined as above: (1) σ ess ( L Ω ) = σ ess ( e L ) . (2) λ ∈ σ ess (cid:0)e L − (cid:1) \ { } if and only if λ = 0 and λ − ∈ σ ess ( L Ω ) .Proof. Since e L = L Ω ⊕ L K , then σ ( e L ) = σ ( L Ω ) ∪ σ ( L K ). But K is compact and hasa Lipschitz boundary, hence L K has a compact resolvent by [13, Theorem 1 . . . σ ess ( L K ) = ∅ and the first assertion in the lemma above follows. Moreover, L Ω and L K are both strictly positive by hypothesis, hence 0 σ ( e L ). It is straightforward that σ ess ( e L ) = n λ ∈ R \ { } : λ − ∈ σ ess (cid:0)e L − (cid:1)o . (cid:3) Essential spectrum of L Ω . With the operator e L introduced above, we canview L Ω as a perturbation of the Landau Hamiltonian L in R d introduced in (2.1).Actually, we define V : L ( R d ) → L ( R d ) as V = e L − − L − . Then we have the following result on the operator V . Lemma 2.5.
The operator V is positive and compact. Moreover, for all f, g ∈ L ( R d ) , h f, V g i L ( R d ) = Z ∂ Ω ∂ R u · ( v Ω − v K ) dS , (2.5) where u = L − f and v = e L − g .Proof. Notice that the form domain H A ( R d ) of L is included in that of e L , andthat for u ∈ H A ( R d ), we have e l ( u ) = Z R d | ( ∇ − ib A ) u | dx . Invoking Lemma 2.1, we get that the operator V is positive.Let us establish the identity in (2.5). Set f = Lu and g = e Lv = L Ω v Ω ⊕ L K v K .Then h f, V g i L ( R d ) = Z Ω Lu · v Ω dx + Z K Lu · v K dx − Z Ω u · L Ω v Ω dx − Z K u · L K v K dx . The identity in (2.5) then follows by integration by parts and by using the boundaryconditions ∂ R v Ω = ∂ R v K = 0.Knowing that the zeroth order trace operators H s ( K ) → H s ( ∂ Ω) and H s loc (Ω) → H s ( ∂ Ω) are compact whenever s > s + 1 / > /
2, cf. [19, Theorem 9 .
4, Chapter1], we conclude from (2.5) that V is a compact operator. (cid:3) The localization of V to the boundary carried out in Lemma 2.5 is by nowa common triviality used in studying boundary value problems, but a sensationaround the time of its invention by Birman, see more in [4, 5]. Theorem 1.1 followsas a corollary of Lemma 2.5: Corollary 2.6.
It holds that σ ess ( L Ω ) = (cid:8) Λ q : q ∈ N \ { } (cid:9) , and for all ε ∈ (0 , b ) and q ∈ N \ { } , σ ( L Ω ) ∩ (Λ q , Λ q + ε ) is finite. MAGNUS GOFFENG, AYMAN KACHMAR, AND MIKAEL PERSSON SUNDQVIST
Proof.
Invoking Lemma 2.4, it suffices to prove that σ ess (cid:0)e L − (cid:1) \ { } = (cid:8) Λ − q : q ∈ N \ { } (cid:9) in order to get the result concerning the essential spectrum of L Ω . Notice that e L − = L − + V with V a compact operator. Hence by Weyl’s theorem, σ ess (cid:0)e L − (cid:1) = σ ess ( L − ). But we know from Section 2.2 that σ ess ( L − ) \{ } = (cid:8) Λ − q : q ∈ N \{ } (cid:9) as was required to prove.Since the operator V is compact and positive, invoking Lemma 2.2, we get that σ (cid:0)e L − (cid:1) ∩ (Λ − q − ε, Λ − q ) is finite. This implies that σ ( L Ω ) ∩ (Λ q , Λ q + ε ) is finite. (cid:3) Proof of Theorem 1.2
In this section we prove Theorem 1.2. The main idea of the proof is, as mentionedabove, to reduce the spectral asymptotics in (1.8) and (1.9) to a similar asymptoticsfor Toeplitz operators on the Landau levels from [9, 20]. The reduction to Toeplitzoperators is by means of localizing to the boundary. The localization to the bound-ary is carried out using the identity (2.5).3.1.
The spectrum of certain Toeplitz operators.
For q ∈ N \ { } and ameasurable set U ⊂ R d the Toeplitz operator S Uq is defined by S Uq = P q χ U P q in L ( R d ) . (3.1)Here χ U is the characteristic function of U . If U is bounded, Arzela-Ascoli’s theoremimplies that χ U P q is a compact operator, because Cauchy estimates for holomor-phic functions can be generalized to the Landau levels. In particular, the Toeplitzoperator S Uq is compact. We state the following deep results on these Toeplitzoperators. Theorem 3.1 ([9, Lemma 3.2]) . Assume that U ⊆ R is a bounded domain withLipschitz boundary. Given q ∈ N \ { } , denote by s ( q )1 ≥ s ( q )2 ≥ . . . the decreasingsequence of eigenvalues of S Uq . Then, lim j → + ∞ (cid:0) j ! s ( q ) j (cid:1) /j = b U )) . Theorem 3.2 ([20, Proposition 7.1]) . Assume that U ⊆ R d is a bounded domain.Given q ∈ N , we let n (cid:0) ε, S Uq (cid:1) denote the number of eigenvalues of S Uq greater than ε . Then n (cid:0) ε, S Uq (cid:1) ∼ (cid:18) q + d − d − (cid:19) d ! (cid:18) | log ε | log | log ε | (cid:19) d as ε → + . The reader will recognize the structure of these results and notice that our mainresults look very much like them. Indeed, our main task will be to reduce oursituation so that these results can be applied.3.2.
The resolvent of the Landau Hamiltonian.
Since L is strictly positive, L − is a bounded operator in L ( R d ) with range D ( L ). Furthermore, L − is anoperator with an integral kernel that we denote by G . This integral kernel iswell-known (see [24]) to be G ( z, ζ ) = 2 b d − (4 π ) d e b ( z · ζ − ζ · z ) / I (cid:18) b | z − ζ | (cid:19) , (3.2)where I ( s ) = Z + ∞ e − s coth( t ) sinh d ( t ) dt. LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 7
Remark 3.3.
The formula for G ( x, y ) when d = 1 is more commonly known: G ( x, y ) = Z + ∞ b π sinh( bs ) exp (cid:16) ib x ∧ y − b bs ) | x − y | (cid:17) ds, (3.3)where x ∧ y = x y − x y = ( z · ζ − ζ · z ) / z = x + ix , ζ = y + iy . Lemma 3.4. L − is an integral operator with kernel G ( z, ζ ) that has the followingsingularity at the diagonal z = ζ : There exist a j ∈ C ∞ ( R d × R d ) , for j ∈ N \ { } ,and b j ∈ C ∞ ( R d × R d ) , for j ∈ N \ { } with j ≥ max(1 , d − , such that for anylarge enough N , there exists a function ˜ a N +1 ∈ C N ( R d × R d ) such that G ( z, ζ ) = 12 π log (cid:0) | z − ζ | (cid:1) + N X j =1 a j ( z, ζ ) | z − ζ | j + N X j =1 b j ( z, ζ ) | z − ζ | j log | z − ζ | + ˜ a N ( z, z − ζ ) , d = 1 G ( z, ζ ) = Γ( d − π d | z − ζ | − d + N X j =1 a j ( z, w ) | z − ζ | − d +2 j (3.4)+ N X j = d − b j ( z, ζ ) | z − ζ | − d +2 j log | z − ζ | + ˜ a N ( z, z − ζ ) , d > . The corresponding expansions, obtained by the term-wise differentiation, exist alsofor ∂ N G ( z, w ) whenever z, w ∈ ∂ Ω . Moreover, G ( z, w ) decays as a Gaussian as | z − w | → + ∞ uniformly in both z and w . The proof of this Lemma is of a computational nature and is deferred to Appen-dix A, where also the asymptotic expansion is computed explicitly.
Remark 3.5.
For d = 1 the integral (3.3) can be expressed in terms of the Whit-taker function (see [8, Section 4.9, formula (31)] and [7, Chapter 6]) as G ( x, y ) = π / b (cid:16) b | x − y | (cid:17) − / exp (cid:16) ib x ∧ y (cid:17) × h W , − (cid:16) b | x − y | (cid:17) + 12 W − , − (cid:16) b | x − y | (cid:17)i . Lemma 3.4 follows in this case from asymptotic formulae for Whittaker functions [7].3.3.
Boundary layer operators.
Recall that K ⊂ R d has been assumed to bea compact subset of R d with smooth boundary and that we defined the domainΩ = R d \ K . Since Ω and K are complementary, the Hilbert space L (cid:0) R d (cid:1) isnaturally decomposed as the orthogonal direct sum L (Ω) ⊕ L ( K ) in the sensethat any function u ∈ L ( R d ) can be uniquely represented as u Ω ⊕ u K where u Ω and u K are the restrictions of u to Ω and K respectively.We let Ψ ∗ ( ∂ Ω) denote the filtered algebra of classical ∗ pseudo-differential opera-tors on the common boundary ∂ Ω of Ω and K . For a reference on pseudo-differentialoperators, the reader is referred to [3, Chapter 5], [15, Chapter 18] or [23, ChapterI]. We fix a classical pseudo-differential operator τ ∈ Ψ ( ∂ Ω). The proofs also worksfor τ ∈ Ψ t ( ∂ Ω), for t < τ is self-adjoint, in the following two subsections thisassumption is needed merely to simplify proofs.For the boundary considerations of this Section, more care in the analysis ofthe Robin boundary differential expressions on ∂ Ω defined in (1.4) is needed. We ∗ Also known under the name 1-step polyhomogeneous.
MAGNUS GOFFENG, AYMAN KACHMAR, AND MIKAEL PERSSON SUNDQVIST note that by [19, Theorem 9.4, Chapter 1] the magnetic normal derivative, cf. (1.4),gives well defined continuous operators for s ∈ R \ { Z + } such that s > / ∂ N, Ω : H s loc (Ω) → H s − / ( ∂ Ω) and ∂ N,K : H s ( K ) → H s − / ( ∂ Ω) . We remind the reader that the normal derivative appearing in ∂ N,K and ∂ N, Ω areboth with respect to the unit outward normal vector to the boundary of Ω. Againfollowing [19, Theorem 9.4, Chapter 1], the trace operators γ , Ω : H s loc (Ω) → H s − / ( ∂ Ω) and γ ,K : H s ( K ) → H s − / ( ∂ Ω) , mapping a function to its boundary value, are continuous for s ∈ R \ { Z + } suchthat s > /
2. For s ∈ R \ Z + such that s > / ∂ Ω, following the expressions (1.4), by means of ∂ R, Ω := ∂ N, Ω + τ γ , Ω : H s loc (Ω) → H s − / ( ∂ Ω) and ∂ R,K := ∂ N,K + τ γ ,K : H s ( K ) → H s − / ( ∂ Ω) . (3.5)As a rule, we suppress the K and the Ω from the notation in these operatorswhenever the domain is clear from the context. We sometimes write ( ∂ R ) x in orderto stress that the differentiation in (3.3) is with respect to the variable x .With G ( x, y ) as in (3.2), we define the operators A , B , A and B , acting onfunctions defined on ∂ Ω, as A u ( x ) = Z ∂ Ω G ( x, y ) u ( y ) dS ( y ) , x ∈ R d \ ∂ Ω , B u ( x ) = Z ∂ Ω (cid:2) ( ∂ N ) y G ( x, y ) (cid:3) u ( y ) dS ( y ) , x ∈ R d \ ∂ Ω ,Au ( x ) = Z ∂ Ω G ( x, y ) u ( y ) dS ( y ) , x ∈ ∂ Ω ,Bu ( x ) = Z ∂ Ω (cid:2) ( ∂ N ) y G ( x, y ) (cid:3) u ( y ) dS ( y ) , x ∈ ∂ Ω . (3.6)The potentials A and B are usually called the single and double layer potentials.They satisfy L A u ( x ) = 0 and L B u ( x ) = 0 in R d \ ∂ Ω. We will write limit relationsat the boundary for these potentials. We refer to [2, Chapter 3, Section 12] wherethe corresponding potentials are considered for the Helmholtz operator, see also[3, Section 5.7] for the low-dimensional case. Since, according to Lemma 3.4, theGreen functions for both L and the Helmholtz operator are globally estimatedpseudo-differential operators with the same asymptotics in the leading terms as x − y →
0, the limit relations in [3] apply here as well. For all x ∈ ∂ Ω it holds thatlim x ∈ K ◦ x → x ( A u )( x ) = ( Au )( x ) , lim x ∈ K ◦ x → x ( B u )( x ) = 12 u ( x ) + ( Bu )( x ) , lim x ∈ Ω x → x ( A u )( x ) = ( Au )( x ) , lim x ∈ Ω x → x ( B u )( x ) = − u ( x ) + ( Bu )( x ) , lim x ∈ Ω x → x ∂ N ( A u )( x ) − lim x ∈ K ◦ x → x ∂ N ( A u )( x ) = u ( x ) . (3.7)We recall from [2, 3] that any function u = u Ω ⊕ u K ∈ H s loc (Ω) ⊕ H s ( K ) expo-nentially decaying at ∞ and solving Lu = 0 in Ω ˙ ∪ K ◦ admits the representation bythe formulas u = B ( γ ,K u K ) − A (cid:0) ∂ N,K u K (cid:1) , in K ◦ , u = A (cid:0) ∂ N, Ω u Ω ) − B ( γ , Ω u Ω ) , in Ω. LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 9
Using the limit values (3.7), we obtain the following formulas connecting Dirichletand Robin data at the boundary ∂ Ω: (cid:16) B + Aτ − (cid:17) γ ,K u K = A ( ∂ R,K u K ) , (cid:16) B + Aτ + 12 (cid:17) γ , Ω u Ω = A ( ∂ R, Ω u Ω ) . (3.8)We will use these relations to define the Dirichlet to Robin and Robin to Dirichletoperators in Section 3.4 below. Before doing so, we present some results on theoperators A and B . Recall that for p ≥
1, the symmetrically normed ideal ofweak Schatten class operators J p,w ( L ( ∂ Ω)) consists of those compact operators C on L ( ∂ Ω) whose singular values { µ k ( C ) } k ∈ N behaves like µ k ( C ) = O ( k − /p ) as k → ∞ . See more in [25]. Lemma 3.6.
The operators A and B are classical pseudo-differential operators oforder − . Furthermore: (1) A defines a self-adjoint operator on L ( ∂ Ω) ; (2) A is elliptic with constant principal symbol; (3) A defines a Fredholm operator A : L ( ∂ Ω) → H ( ∂ Ω) whose index vanishes; (4) A and B , considered as operators on L (Ω) , belong to the weak Schattenclass J d − ,w ( L ( ∂ Ω)) .Proof.
It follows directly from the asymptotics in Lemma 3.4 that A and B areclassical pseudo-differential operators of order −
1, see for instance [15, Theorem18.2.8]. As such, the Weyl law on ∂ Ω implies that A and B belong to the weakSchatten class J d − ,w ( L ( ∂ Ω)). It is also clear from Lemma 3.4 that σ − ( A ) = Γ( d − π d . Hence A is elliptic. Furthermore, Equation (3.2) implies that G ( z, w ) = G ( w, z ) so A is self-adjoint on L ( ∂ Ω). Since σ − ( A ) is a constant mapping, itis a lower order perturbation of an invertible pseudo-differential operator and thestatement ind( A : L ( ∂ Ω) → H ( ∂ Ω)) = 0 follows. (cid:3)
Lemma 3.7.
The elliptic operator A defines an isomorphism A : L ( ∂ Ω) → H ( ∂ Ω) .Proof. To prove that A is an isomorphism, we follow the proof of [26, Chapter 7,Proposition 11.5] with the necessary modifications. Since the index of A : L ( ∂ Ω) → H ( ∂ Ω) vanishes, it suffices to prove that the operator A is injective. By ellipticregularity, it suffices to prove that A : C ∞ ( ∂ Ω) → C ∞ ( ∂ Ω) is injective.Assume that h ∈ C ∞ ( ∂ Ω) with Ah = 0. If we define u ∈ C ∞ ( K ◦ ) by u ( x ) = A h ( x ), x ∈ K ◦ , then u satisfies ( − ( ∇ − ib A ) u = 0 in K ◦ ,u = 0 on ∂ Ω.We use (2.3) and integrate by parts, to get0 = (cid:10) − ( ∇ − b A ) u, u (cid:11) L ( K ) = bd k u k L ( K ) + d X j =1 k Q j u k L ( K ) . This implies that u ≡ K , i.e. A h ( x ) ≡ K ◦ . (3.9)It follows from the limit relations (3.7) that ∂ N ( A h )( x ) makes a jump across theboundary ∂ Ω of size h , so if we let w ( x ) = A h ( x ), x ∈ Ω, then it satisfies ( − ( ∇ − ib A ) w = 0 in Ω ,∂ N w = h on ∂ Ω. (3.10)
Since, again by (3.7), A h does not jump across ∂ Ω, we see by (3.9) that w = 0 on ∂ Ω.From the exponential decay of G ( x, y ) as | x − y | → + ∞ it follows that w ( x ) = O ( | x | − N ) as | x | → + ∞ for all N >
0. This also applies to all derivatives of w .Moreover w is smooth. Hence we can integrate by parts in Ω to find0 = (cid:10) − ( ∇ − ib A ) w, w (cid:11) L (Ω) = bd k w k L (Ω) + d X j =1 k Q j w k L (Ω) , and hence w ≡ h = 0 on ∂ Ω. (cid:3) The Dirichlet to Robin and Robin to Dirichlet operators.
Let ϕ ∈ L ( ∂ Ω) be given, and let u be a solution with exponential decay at infinity to theexterior Robin problem ( Lu = 0 , in Ω ,∂ R u = ϕ, on ∂ Ω . We will see below in Equation (3.11) that the existence of u is guaranteed in a sub-space of finite codimension. This solution is unique, provided certain orthogonalityconditions are imposed. We denote by T R → D Ω ϕ the boundary values of u at ∂ Ω,whenever ϕ admits a solution u . The operator T R → D Ω : ϕ T R → D Ω ϕ is called the exterior Robin to Dirichlet operator for the differential equation Lu = 0. We de-fine the interior Robin to Dirichlet operator T R → DK in a similar way. Their inverseoperators, associating Robin data of solutions to their Dirichlet data are calledthe exterior and interior Dirichlet to Robin operators, and are denoted T D → R Ω and T D → RK respectively.Using the relations in (3.8) we find that these operators in fact are independenton the choice of solution u for ϕ outside a finite-dimensional subspace. It followsfrom (3.8) that: (cid:16) B + Aτ − (cid:17) T R → DK = A, on A − (cid:16) B + Aτ − (cid:17) C ∞ ( ∂ Ω); (cid:16) B + Aτ + 12 (cid:17) T R → D Ω = A, on A − (cid:16) B + Aτ + 12 (cid:17) C ∞ ( ∂ Ω); T D → RK = A − (cid:16) B + Aτ − (cid:17) ; T D → R Ω = A − (cid:16) B + Aτ + 12 (cid:17) . (3.11)These equations determine T R → DK , T R → D Ω , T D → RK and T D → R Ω outside a finite-dimensional subspace while B + Aτ ± are elliptic pseudo-differential operatorsof order 0. After a choice of extension, Equation (3.11) allows us to consider theoperators T R → DK , T R → D Ω , T D → RK and T D → R Ω as pseudo-differential operators on ∂ Ω. More precisely, we have the following standard result for Dirichlet to Robinoperators. The proof can be found in [26, Appendix C of Chapter 12].
Proposition 3.8.
The interior and exterior Robin to Dirichlet and Dirichlet toRobin operators are given by the relations in (3.11) , and are elliptic pseudodiffer-ential operators with constant principal symbols: T R → DK , T R → D Ω ∈ Ψ − ( ∂ Ω) and T D → RK , T D → R Ω ∈ Ψ ( ∂ Ω) . Very oftenly, Equation (3.11) determines the Dirichlet to Robin operators; theoperators B + Aτ ± are, in a sense made precise below, generically invertible.This fact is based on the following lemma. LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 11
Lemma 3.9.
Assume that M is a closed manifold and that s > . Let T ∈ Ψ ( M ) be an elliptic operator of index and A ∈ Ψ − s ( M ) be an invertible operator A : L ( M ) → H s ( M ) . Viewing A − and A − T as unbounded operators on L ( M ) ,then for any ε / ∈ − σ ( A − T ) , the bounded operator T + εA : L ( M ) → L ( M ) is invertible. The statement of the Lemma makes sense because T is a pseudo-differentialoperator of order zero and as such it preserves the domain H s ( M ) of A − . We alsonote that by elliptic regularity, the set − σ ( A − T ) is the same when changing thedomain and range of A − T to H t ( M ) for any t . Proof.
As ind( T ) = 0 and A is of negative order, the operator T + εA is an ellipticpseudo-differential operator of order 0. As such, it defines an operator H t ( M ) → H t ( M ) with index 0 for any ε and t . Hence T + εA is invertible if and only ifker( T + εA ) = 0. However, A is invertible so ker( T + εA ) = 0 holds if and onlyif ker( A − T + ε ) = 0. By elliptic regularity these subspaces do not depend on thechoice of domain in the Sobolev scale. The operator A − T is an elliptic pseudo-differential operator of order s with index 0. It follows that ind( A − T + ε ) = 0.Thus ker( A − T + ε ) = 0 holds if and only if − ε / ∈ σ ( A − T ). (cid:3) We say that τ is generic if the operators T ± ,τ := B + Aτ ± D of order m is invertible as an operator between Sobolev spaces D : H s ( ∂ Ω) → H s − m ( ∂ Ω), for some s ∈ R , if and only if D is invertible inside the algebra ofpseudo-differential operators. The following Corollary motivates the terminology generic . Corollary 3.10.
Letting τ , A and B be as above. For all ε ∈ [ − , outside afinite subset, τ + ε is generic.Proof. By [23, Theorem I.8 . σ ( A − T ± ,τ ) is not equal to C , itis a discrete subset of C and [ − , ∩ σ ( − A − T + ,τ ) ∩ σ ( − A − T − ,τ ) is a finite set.The Corollary follows from Lemma 3.9 provided there exists λ ± ∈ C such that A − T ± ,τ + λ ± are invertible. We note that the principal symbols ± σ ( A − T ± ,τ )are positive constant functions on S ∗ ∂ Ω. Existence of λ ± ∈ C follows from theGårding inequality ([15, Theorem 18 . . (cid:3) We turn our attention to associating Robin data on ∂ Ω to Dirichlet data forfunctions in the Landau subspace L q , or more generally to solutions of the homo-geneous equation ( L − Λ q ) u = 0 in K ◦ . The construction is well known and can befound in, for instance, [26, Chapter 7.12 and Appendix C of Chapter 12].Let Q Kq ⊆ C ∞ ( K ) denote the space of solutions u ∈ C ∞ ( K ) to ( L − Λ q ) u = 0in K ◦ and let Q ∂ Ω q ⊆ C ∞ ( ∂ Ω) be the image of Q Kq under the restriction mapping C ∞ ( K ) → C ∞ ( ∂ Ω). Since L − Λ q is a strongly elliptic operator in K , the space Q ∂ Ω q ⊆ C ∞ ( ∂ Ω) has finite codimension. Since the eigenvalue multiplicities of L in K equipped with Dirichlet conditions on ∂ Ω are finite, the kernel of Q Kq → Q ∂ Ω q isfinite-dimensional.This means that we can, for any function ϕ ∈ Q ∂ Ω q , solve ( ( L − Λ q ) u = 0 , in K ◦ , u = ϕ, on ∂ Ω. The condition that u is L -orthogonal to ker (cid:0) Q Kq → Q ∂ Ω q (cid:1) guarantees a uniquesolution. Define the corresponding solution operator T D → Rq : Q ∂ Ω q → C ∞ ( ∂ Ω) , T D → Rq ϕ := ∂ R,K u. We let T D → Rq : C ∞ ( ∂ Ω) → C ∞ ( ∂ Ω) denote any extension of this operator, whichexists since Q ∂ Ω q ⊆ C ∞ ( ∂ Ω) has finite codimension. The following result is standardand follows from [26, Chapter 7].
Lemma 3.11.
The operator T D → Rq possesses the following properties: (1) T D → Rq is an elliptic pseudo-differential operator of order with constantpositive principal symbol. (2) For any s , ind (cid:0) T D → Rq : H s ( ∂ Ω) → H s − ( ∂ Ω) (cid:1) = 0 . (3) There exists a number c = c ( ∂ Ω , q ) ∈ R such that T D → Rq : H / ( ∂ Ω) → H − / ( ∂ Ω) is invertible as an operator on L ( ∂ Ω) as long as τ ≥ c .Proof. The computation of the principal symbol of T D → Rq can be found in Proposi-tion C.1 of [26, Chapter 12]. The identity ind (cid:0) T D → Rq : H s ( ∂ Ω) → H s − ( ∂ Ω) (cid:1) = 0follows in the same way as in the proof of Lemma 3.6. From the index computationfor T D → Rq we conclude that T D → Rq is an isomorphism if and only if T D → Rq is injec-tive. It follows from the Gårding inequality that there exists a positive constant c such that for τ ≥
0, for a certain constant c ′ τ > (cid:10) T D → Rq u, u (cid:11) ≥ c ′ τ k u k H / ( ∂ Ω) − c k u k L ( ∂ Ω) . Hence, as long as τ ≥ c , we have Re h T D → Rq u, u i ≥ c ′ τ − c k u k H / ( ∂ Ω) . Injectivity of T D → Rq : H / ( ∂ Ω) → H − / ( ∂ Ω) assuming τ ≥ c follows. (cid:3) Remark 3.12.
We note that by construction, T D → Rq coincides with ∂ R,K outside afinite-dimensional subspace. Hence, for q ∈ N \{ } , a finite rank smoothing operator S q ∈ Ψ −∞ ( ∂ Ω) exists, such that, as long as u ∈ C ∞ ( K ) satisfies ( L − Λ q ) u = 0 in K ◦ , then ∂ R,K u = ( T D → Rq + S q ) γ ,K u. Reduction to a Toeplitz operator.
In this subsection we prove Theo-rem 1.2 modulo a technical Lemma that we prove in the next section. We assume asabove that K ⊂ R d is compact with smooth boundary upon which τ is a classicalpseudo-differential operator which we for simplicity assume to have order 0. In theprevious subsections, we made the assumption that τ was self-adjoint to simplifyproofs while in this subsection it is necessary for the results to hold. Let q ∈ N \ { } and pick δ > (cid:16)(cid:0) Λ − q − δ, Λ − q + 2 δ (cid:1) \ (cid:8) Λ − q (cid:9)(cid:17) ∩ σ ess ( e L − ) = ∅ . Denote by (cid:8) r ( q ) j (cid:9) j ≥ the decreasing sequence of eigenvalues of e L − in the interval(Λ − q , Λ − q + δ ). For each q ∈ N \ { } we introduce the operator T q = P q V P q , (3.12)where, as before, P q is the orthogonal projection onto the eigenspace L q associatedwith Λ q and V = e L − − L − . By Lemma 2.5, V is a positive and compact operator.These properties are inherited by T q . Denote by { t ( q ) j } the decreasing sequence ofeigenvalues of T q . The next lemma, proved in [22, Proposition 2.2], shows that r ( q ) j − Λ − q are close to the eigenvalues of T q . LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 13
Lemma 3.13 ([22, Proposition 2.2]) . Given ε > there exist integers l and j such that (1 − ε ) t ( q ) j + l ≤ r ( q ) j − Λ − q ≤ (1 + ε ) t ( q ) j − l , ∀ j ≥ j . The spectrum of T q will be related further to the spectrums of Toeplitz oper-ators for generic operators τ . Recall that given a compact domain U ⊂ R d , weintroduced in (3.1) the Toeplitz operator S Uq . We will prove now the followingresult. Lemma 3.14.
For all q ∈ N \ { } there exists a finite-dimensional subspace W q ⊆L q such that if K ⊂ K ⊂ K are compact domains with ∂K i ∩ ∂K = ∅ (for i = 0 and i = 1 ) there exists a constant C > such that C h f, S K q f i L ( R d ) ≤ h f, T q f i L ( R d ) ≤ C h f, S K q f i L ( R d ) ∀ f ∈ L q ⊖W q . (3.13)The proof of Lemma 3.14 is by reduction of the operator T q to a pseudo-differentialoperator on the common boundary ∂ Ω of Ω and K . We postpone the proof to Sec-tion 4 below, and continue instead with the proof of (1.8). Corollary 3.15.
Whenever K ⊂ R is compact with C ∞ -boundary, lim j → + ∞ (cid:16) j ! (cid:0) r ( q ) j − Λ − q (cid:1)(cid:17) /j = b (cid:0) Cap( K ) (cid:1) . In particular, (1.8) holds true.Proof.
Invoking the variational min-max principle, the result of Lemma 3.14 pro-vides us with a sufficiently large integer j ∈ N such that, for all j ≥ j , we have,1 C s ( q ) j,K ≤ t ( q ) j ≤ Cs ( q ) j,K . Here { s ( q ) j,K } j and { s ( q ) j,K } j are the decreasing sequences of eigenvalues of S K q and S K q respectively. Applying the result of Theorem 3.1 in the inequality above, weget b K )) ≤ lim j → + ∞ (cid:16) j ! t ( q ) j (cid:17) /j ≤ b K )) . Since both K ⊂ K ⊂ K are arbitrary, we get by making them close to K ,lim j → + ∞ (cid:16) j ! t ( q ) j (cid:17) /j = b K )) . Applying the above asymptotic limit in the estimate of Lemma 3.13, we get theannounced result in Corollary 3.15 above. (cid:3)
Corollary 3.16.
Equation (1.9) holds true.Proof.
This is clear from Lemma 3.14 and Theorem 3.2. (cid:3)
Summing up the results of Corollaries 2.6, 3.15 and 3.16, we end up with theproof of Theorem 1.2. All that remains is to prove Lemma 3.14. That will be thesubject of the next section. 4.
Proof of Lemma 3.14
The aim of this section is to prove Lemma 3.14. The proof in this subsectiongoes along similar lines as in Subsection 4.2 of [21]. Recall the operators A and B from (3.6). Lemma 4.1.
Consider the elliptic operators T ± ,τ = B + Aτ ± ∈ Ψ ( ∂ Ω) . Thereexist elliptic operators R ± ,τ ∈ Ψ ( ∂ Ω) , with principal symbol σ ( R ± ,τ ) = ± , suchthat the operators T ± ,τ R ± ,τ − ∈ Ψ −∞ ( ∂ Ω) and R ± ,τ T ± ,τ − ∈ Ψ −∞ ( ∂ Ω) are of finite rank. Lemma 4.1 could be considered folklore. In lack of a reference we provide a proofof its statement.
Proof.
We let Ψ −∞ fin ( ∂ Ω) ⊆ Ψ ( ∂ Ω) denote the ideal of finite rank smoothing oper-ators. We consider the unital algebras A fin := Ψ ( ∂ Ω) / Ψ −∞ fin ( ∂ Ω) and A := Ψ ( ∂ Ω) / Ψ −∞ ( ∂ Ω) . There is a quotient mapping A fin → A whose kernel is Ψ −∞ ( ∂ Ω) / Ψ −∞ fin ( ∂ Ω). Itfollows by means of the standard techniques of pseudo-differential operators that anelement T ∈ Ψ ( ∂ Ω) is elliptic if and only if the equivalence class T mod Ψ −∞ ( ∂ Ω)is an invertible element of A . Hence the Lemma follows if we can prove that if˜ a ∈ A fin satisfies that a := ˜ a mod Ψ −∞ ( ∂ Ω) / Ψ −∞ fin ( ∂ Ω) ∈ A is invertible, then sois ˜ a .We choose a lift ˜ r ∈ A fin of a − and consider the elements in Ψ −∞ ( ∂ Ω) / Ψ −∞ fin ( ∂ Ω)defined by s L := ˜ r ˜ a − A fin and s R := ˜ a ˜ r − A fin . We can lift s L and s R to smoothing operators S L , S R ∈ Ψ −∞ ( ∂ Ω). The operators1 + S L and 1 + S R are elliptic operators of index 0. Fredholm operators of index0 are invertible modulo finite rank operators. If follows by elliptic regularity that1 + S L and 1 + S R are invertible modulo finite rank smoothing operators. Hence1 A fin + s L and 1 A fin + s R are invertible elements of A fin .Let us define ˜ r R := ˜ r (1 A fin + s R ) − and ˜ r L := (1 A fin + s L ) − ˜ r . A direct compu-tation shows that ˜ a ˜ r R = ˜ r L ˜ a = 1 A fin . Multiplying the identity ˜ a ˜ r R = 1 A fin with ˜ r L from the left proves that ˜ r R = ˜ r L . Itfollows that ˜ a is invertible with inverse ˜ r L . (cid:3) By Lemma 3.9, it is generically the case that T ± ,τ is invertible. To simplifynotation we set ˆ F ± ,τ := R ± ,τ T ± ,τ − Lemma 4.2.
Let q ∈ N \ { } . There exists a finite rank smoothing operator F q : L ( R d ) → L ( ∂ Ω) such that for all f, g ∈ L ( R d ) , h f, T q g i L ( R d ) = 1Λ q Z ∂ Ω ( P q f ) · T τ,q ( P q g ) dS + h γ P q f, F q g i L ( ∂ Ω) , (4.1) where T τ,q is the elliptic operator defined by T τ,q := (cid:0) T D → Rq + S q (cid:1) ∗ ( R − ,τ − R + ,τ ) A (cid:0) T D → Rq + S q (cid:1) ∈ Ψ ( ∂ Ω) . Proof.
We set u = L − P q f = Λ − q P q f , v = e L − P q g = v Ω ⊕ v K and w = L − P q g =Λ − q P q g . Notice that h f, T q g i L ( R d ) = h P q f, V P q g i L ( R d ) where V is the operator defined in (2.5). Invoking Lemma 2.5, we write, h P q f, V P q g i L ( R d ) = Z ∂ Ω ∂ R u · ( v Ω − v K ) dS = Z ∂ Ω ∂ R u · ( v Ω − w + w − v K ) dS. LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 15
Note that since u ∈ H A ( R d ), ∂ R u = ∂ R,K u = ∂ R, Ω u . Using (3.11) and Lemma 4.1,we can write further, h P q f, V P q g i L ( R d ) = Z ∂ Ω ∂ R u · ( R + ,τ A ( ∂ R, Ω ( v Ω − w )) + R − ,τ A ( ∂ R,K ( w − v K ))) dS (4.2) − Z ∂ Ω ∂ R u · ˆ F + ,τ ( v Ω − w ) + ˆ F − ,τ ( w − v K )) dS. Notice that v Ω and v K are in the domain of the operators L Ω and L K respectively,hence ∂ R v Ω = ∂ R v K = 0. By construction, L q u = L q w = 0 in K , and Remark 3.12implies that ∂ R u = ( T D → Rq + S q ) γ u and ∂ R w = ( T D → Rq + S q ) γ w, (4.3)for some finite rank smoothing operator S q ∈ Ψ −∞ ( ∂ Ω). Consequently, after ap-plying this identity to the first term in (4.2) we get h P q f,V P q g i L ( R d ) + Z ∂ Ω ∂ R u · ˆ F + ,τ ( v Ω − w ) + ˆ F − ,τ ( w − v K )) dS = Z ∂ Ω ( T D → Rq + S q ) u · ( R − ,τ − R + ,τ ) A ( T D → Rq + S q ) w dS = h u, T τ,q w i L ( ∂ Ω) . As for the second term in (4.2), Z ∂ Ω ∂ R u · ˆ F + ,τ ( v Ω − w ) + ˆ F − ,τ ( w − v K )) dS = Z ∂ Ω u · ( T D → Rq + S q ) ∗ (Λ − q ( ˆ F − ,τ − ˆ F + ,τ ) + ( ˆ F + ,τ L − − ˆ F − ,τ L − K )) P q | {z } Λ q F q g dS. Since ˆ F ± ,τ are of finite rank, it is clear that F q is of finite rank. (cid:3) Lemma 4.3.
The operator T τ,q has discrete spectrum, and there exists a finite rankoperator S τ,q ∈ Ψ −∞ ( ∂ Ω) such that for some b, C > , k ϕ k H / ( ∂ Ω) ≤ C (cid:0) Re h ϕ, T τ,q ϕ i L ( ∂ Ω) + b k S τ,q ϕ k L ( ∂ Ω) (cid:1) . Proof.
By an argument similar to that of Corollary 3.10, it follows from [23, The-orem I.8 .
4] and the Gårding inequality that T τ,q has discrete spectrum. We definethe elliptic self-adjoint first order pseudo-differential operator˜ T τ,q := 12 ( T τ,q + T ∗ τ,q ) . Since T τ,q has positive principal symbol, we see that T τ,q − ˜ T τ,q ∈ Ψ ( ∂ Ω) definesa bounded operator. The Gårding inequality implies that for some b, C > k ϕ k H / ( ∂ Ω) ≤ C (cid:0) h ϕ, ˜ T τ,q ϕ i L ( ∂ Ω) + b k ϕ k L ( ∂ Ω) (cid:1) . Since ˜ T τ,q is of order 1, elliptic and self-adjoint, its spectrum is a discrete subsetof R . The Gårding inequality implies that ˜ T τ,q is bounded from below, so thespectrum of ˜ T τ,q only accumulates at + ∞ and there are only finitely many non-positive eigenvalues. We define S τ,q ∈ Ψ −∞ ( ∂ Ω) as the finite rank projection ontothe non-positive eigenspace of ˜ T τ,q . For a, possibly new, constant C , the Lemmafollows because Re h ϕ, T τ,q ϕ i L ( ∂ Ω) = h ϕ, ˜ T τ,q ϕ i L ( ∂ Ω) . (cid:3) Proposition 4.4.
The linear operator H ( K ) → H − ( K ) ⊕ H / ( ∂ Ω) , f ( L − Λ q ) f ⊕ γ ,K f, (4.4) is Fredholm.Proof. We note that the strongly elliptic differential operator L − Λ q defines anelliptic boundary value problem when equipped with Dirichlet condition as in (4.4).It follows that the linear operator given in (4.4) is Fredholm because it is a compactperturbation of the operator f ( L + λ ) f ⊕ γ ,K f which is invertible for ℜ ( λ ) largeenough. (cid:3) Remark 4.5.
As a consequence of Proposition 4.4, there exists a bounded linearextension mapping Q q : H / ( ∂ Ω) → H ( K ) satisfying γ Q q ϕ = ϕ and ( L − Λ q ) Q q ϕ = 0 in K ◦ for ϕ outside a finite-dimensional subspace of H / ( ∂ Ω). Thisfinite-dimensional subspace can be chosen as the orthogonal complement of thespace γ ,K (cid:0) ker (cid:0) ( L − Λ q ) : H ( K ) → H − ( K ) (cid:1)(cid:1) . We conclude that there exists afinite rank operator F K on H ( K ), smoothing in the interior, such that whenever f ∈ H ( K ) satisfies ( L − Λ q ) f = 0 in K ◦ , k f k H ( K ) ≤ C (cid:0) k γ f k H / ( ∂ Ω) + k F K f k H ( K ) (cid:1) , for some constant C . Lemma 4.6.
Let q ∈ N \ { } and let F q : L ( R d ) → L ( ∂ Ω) be the finite rankoperator of Lemma 4.2, S τ,q ∈ Ψ −∞ ( ∂ Ω) the finite rank operator of Lemma 4.3 and P q the orthogonal projection on the Landau level L q . There exist constants b > , C > , such that for all f ∈ L ( R d ) it holds that, C k γ P q f k H / ( ∂ Ω) ( k γ P q f k H / ( ∂ Ω) − k F q f k L ( ∂ Ω) − b k S τ,q γ P q f k L ( ∂ Ω) ) ≤ h f, T q f i L ( R d ) ≤ C k γ P q f k H / ( ∂ Ω) ( k γ P q f k H / ( ∂ Ω) + k F q f k L ( ∂ Ω) ) . Proof.
Lemma 3.6 states that the operator T τ,q from Lemma 4.2 is an ellipticpseudo-differential operator of order 1 with positive principal symbol. Hence, byLemma 4.3, there are constants b, C > C k ϕ k H / ( ∂ Ω) ( k ϕ k H / ( ∂ Ω) − b k S τ,q ϕ k L ( ∂ Ω) ) ≤ Re h ϕ, T τ,q ϕ i L ( ∂ Ω) ≤ C k ϕ k H / ( ∂ Ω) , for all ϕ ∈ H / ( ∂ Ω). Applying the above estimates with ϕ = γ P q f and f ∈ L ( R d ), and recalling (4.1), we get that the double inequality announced in theabove lemma holds for all f ∈ L ( R d ) due to Lemma 4.2 and the fact thatRe h ϕ, T τ,q ϕ i L ( ∂ Ω) = h f, T q f i + Re h ϕ, F q f i L ( ∂ Ω) . (cid:3) Proof of Lemma 3.14.
Let S τ,q ∈ Ψ −∞ ( ∂ Ω) denote the finite rank operator fromLemma 4.2. The operator S τ,q γ P q : L ( R d ) → L ( ∂ Ω) is a well defined finiterank operator since S τ,q is finite rank and P q L ( R d ) ⊆ C ∞ ( R d ). Recall the finiterank operator F K on H ( K ) from Remark 4.5 and let F K P q : L ( R d ) → H ( K )denote the finite rank operator f ( P q f ) | K F K [( P q f ) | K ] which is well definedsince ( P q f ) | K ∈ C ∞ ( K ). We define the space W q ⊆ L q by means of L q ⊖ W q := ker ( S τ,q γ P q ) ∩ ker F q ∩ ker F K P q ∩ L q ⊆ L q . The space W q is of finite dimension because all operators S τ,q γ P q , F q and F K P q are of finite rank. Step 1. Lower bound.
LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 17
We prove that the lower bound in (3.13) is valid for all f ∈ W ⊥ q . For simplicitywe set ϕ := γ f . By the definition of T q from (3.12), the estimate of Lemma 4.6gives, h f, T q f i L ( R d ) ≥ C k ϕ k H / ( ∂ Ω) . So it suffices to prove that h f, S Kq f i L ( R d ) ≤ C ′ k ϕ k H / ( ∂ Ω) , for some positive constant C ′ . Recalling the definition of S Kq , and using that k f k L ( K ) ≤ k f k H ( K ) , this follows once showing the estimate k f k H ( K ) ≤ C ′ k ϕ k H / ( ∂ Ω) . (4.5)Since ( L − Λ q ) f = 0 this estimate follows from Remark 4.5. Step 2. Upper bound.
Now we establish the upper bound in (3.13). Let f ∈ L ( R d ) and u = P q f , theprojection of f onto the eigenspace L q . Notice that the trace theorem, [19, Theorem9 .
4, Chapter 1] gives, k γ u k H / ( ∂ Ω) ≤ C k u k H ( K ) , for some positive constant C . Notice that ( L − Λ q ) u = 0. By the elliptic regularity,given a domain K such that K ⊂ K , there exists a constant C K such that, k u k H ( K ) ≤ C K (cid:0) k L q u k L ( K ) + k u k L ( K ) (cid:1) = C K k u k L ( K ) . Summing up, we get, k γ P q f k H / ( ∂ Ω) ≤ C k P q f k L ( K ) , ∀ f ∈ L ( R d ) . Substituting the above inequality in the estimate of Lemma 4.6, we obtain theupper bound announced in (3.13). (cid:3)
Appendix A. Proof of Lemma 3.4
The proof of Lemma 3.4, and the expansion of the function G defined in (3.2),is based on an expansion of the function I . Lemma A.1.
The function I ( s ) := Z + ∞ e − s coth( t ) sinh d ( t ) dt can be written as I ( s ) = I ( s ) + I ∞ ( s ) where (1) I ( s ) , I ∞ ( s ) = O ( e − s ) as s → + ∞ . (2) I ∞ ∈ C ∞ ( R ) . (3) The function I ∈ C ∞ ( R + ) admits an asymptotic expansion for small s : I ( s ) = + ∞ X j =1 − d ( e − sa c j − c ′ j ) s j − + ∞ X j =0 d j s j log( s ) , (A.1) where c j := ⌊ ( d − j ) / ⌋− X k =0 ( − k (cid:18) d − k (cid:19) ( d − k + 1)) j +1 a d − k − − j , − d ≤ j < , + ∞ X k = ⌈ ( d − j − / ⌉− ( − k (cid:18) d − k (cid:19) a d − k − j (2 k + 1 − d ) j +1 , ≤ j ≤ d − , , j > d − .c ′ j := X j ≥ k ≥ j − k − d ≡ ( − ( j + k + d − / a k ( j − k )! k · k ! (cid:18) d − j − k + d − (cid:19) + ( − ( j + d − / j ! (cid:18) d − j + d − (cid:19) ( γ + log( a ) + 1) , j − d ≡ , j ≥ , X j ≥ k ≥ j − k − d ≡ ( − ( j + k + d − / a k ( j − k )! k · k ! (cid:18) d − j − k + d − (cid:19) , j − d ≡ , j ≥ .γ + log( a ) + 1 , j = 0 and d is odd. , j = 0 and d is even. d j := X j ≥ k ≥ j − k − d ≡ ( − ( j + k + d − / a k ( j − k )! k · k ! (cid:18) d − j − k + d − (cid:19) . Here a = coth(1) and γ is the Euler-Mascheroni constant.Proof. We write I ( s ) := Z e − s coth( t ) sinh d ( t ) dt and I ∞ ( s ) := Z + ∞ e − s coth( t ) sinh d ( t ) dt. It is easily verified that I ∞ ∈ C ∞ ( R ). Since coth( t ) ≥ t ≥ I ∞ ( s ) = O ( e − s )as s → + ∞ . Since coth( t ) ≥ / t ∈ [0 , I ( s ) = O ( e − s ) as s → + ∞ follows once one notice that the singularity at t = 0 from the sinh d ( t ) is canceledby e − s coth( t ) / .Let us turn to the unpleasant computation of I . After the change of variables u = coth( t ) the integral defining I transforms to I ( s ) = Z + ∞ a e − su ( u − d − du = Z + ∞ a e − su u d − (cid:16) − u (cid:17) d − du. Here a := coth(1) >
1. Using the Taylor expansion of (cid:0) − u (cid:1) d − , which convergesuniformly for u ≥ coth(1), we arrive at the identity I ( s ) = + ∞ X k =0 ( − k (cid:18) d − k (cid:19) Z + ∞ a e − su u d − k +1) du (A.2)= + ∞ X k =0 ( − k (cid:18) d − k (cid:19) s k +1 − d g d − k +1) ( sa ) . where g m ( t ) := Z + ∞ t e − u u m du. LUSTERS FOR THE MAGNETIC LAPLACIAN WITH ROBIN CONDITION 19
After an integration by parts, one arrives at the identity g m ( t ) = e − t m X j =0 ( m ) j +1 t m − j , m ≥ ,e − t − m − X j =1 t m + j ( − m − j − γ + log( t ) + P + ∞ j =0 ( − j t j j · j ! ( − m − , m < . Here ( m ) l := m ( m − · · · ( m − l + 1) denotes the Pochhammer symbol. Puttingthis into (A.2), I ( s ) = ⌊ ( d − / ⌋ X k =0 d − k +1) X j =0 ( − k (cid:18) d − k (cid:19) ( d − k + 1)) j e − sa a d − k +1) − j s − j − + + ∞ X k = ⌊ d/ ⌋ ( − k (cid:18) d − k (cid:19) e − sa k +1) − d X j =1 a d − k +1) − j (2 k + 1 − d ) j s j − − + ∞ X k = ⌊ d/ ⌋ ( − k (cid:18) d − k (cid:19) s k +1 − d ( γ + log( a ) + 1) + s k +1 − d log( s )(2 k + 1 − d )! − + ∞ X k = ⌊ d/ ⌋ + ∞ X j =1 ( − k + j (cid:18) d − k (cid:19) a j j · j ! s k +1 − d + j (2 k + 1 − d )! . Rearranging these terms leads to the expression (A.1). (cid:3)
Proof of Lemma 3.4.
Using Lemma A.1, we have that G ( z, ζ ) = + ∞ X j =1 − d b j + d − d +2 j − π d e b ( z · ζ − ζ · z ) / (cid:0) e − ab | z − ζ | / c j − c ′ j (cid:1) | z − ζ | j − + ∞ X j =0 d j b j + d − d +2 j − π d e b ( z · ζ − ζ · z ) / | z − ζ | j log (cid:16) b | z − ζ | (cid:17) + 2 b d − (4 π ) d e b ( z · ζ − ζ · z ) / I ∞ (cid:16) b | z − ζ | (cid:17) . From these expressions, the Lemma follows. (cid:3)
Acknowledgements
The authors wish to thank Grigori Rozenblum who suggested the approach weused from the theory of pseudo-differential operators. Part of this work has beenprepared in the Erwin Schrödinger Institute (ESI) - Vienna which is gratefullyacknowledged. The authors MG and MPS also wish to thank the Mittag–Lefflerinstitute for a productive stay in 2012. AK is supported by a grant from LebaneseUniversity.
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E-mail address , M. Goffeng: [email protected]
E-mail address , A. Kachmar: [email protected]
E-mail address , M. Persson Sundqvist:, M. Persson Sundqvist: