Counting function of magnetic resonances for exterior problems
aa r X i v : . [ m a t h . SP ] M a y COUNTING FUNCTION OF MAGNETIC RESONANCES FOR EXTERIOR PROBLEMS
VINCENT BRUNEAU AND DIOMBA SAMBOUA
BSTRACT . We study the asymptotic distribution of the resonances near the Landau levels Λ q =(2 q + 1) b , q ∈ N , of the Dirichlet (resp. Neumann, resp. Robin) realization in the exterior of acompact domain of R of the 3D Schr¨odinger Schr¨odinger operator with constant magnetic field ofscalar intensity b > . We investigate the corresponding resonance counting function and obtain themain asymptotic term. In particular, we prove the accumulation of resonances at the Landau levelsand the existence of resonance free sectors. In some cases, it provides the discreteness of the set ofembedded eigenvalues near the Landau levels.
1. I
NTRODUCTION
It is now well known that perturbations of magnetic Schr¨odinger operators can generate spectralaccumulations near the Landau levels. In the D case, the free Hamiltonian (the Landau Hamiltonian )admits pure point spectrum with eigenvalues (the so called
Landau levels ) of infinite multiplicity. Itsperturbations by an electric potential of definite sign (even if it is compactly supported) produceconcentration of eigenvalues at the Landau levels (see [18], [19], [15], [7]). More recently, similarphenomena are obtained for perturbations by obstacle (see [17] for the Dirichlet problem, [16] for theNeumann problem and [10] for the Robin boundary condition). Let us also mention the work [14]where is considered potential perturbations which are not of fixed sign.The study of the D Schr¨odinger operator is more complicated because the spectrum of the freeHamiltonian is continuous (it is [ b, + ∞ ) where b > is the strength of the constant magnetic field).For perturbations of such operators, the spectral concentration can be analyse on several way. Forexample, it is possible to prove that some axisymmetric perturbations can produce an infinite numberof embedded eigenvalues near the Landau levels (see [5]). In a more general framework it is statedthat the Landau levels are singularities of the Spectral Shift Function (see [8]) and are accumulationpoints of resonances (see [3], [4]). These results are done for a wide class of potentials of definitesign, but it is also important to consider the cases of obstacle perturbations. For example, magneticboundary problems appear in the Ginzgurg-Landau theory of superconductors, in the theory of Bose-Einstein condensat es, and in the study of edge states in Quantum Mechanics (see for instance [6],[12], [1], [9],...).In this paper, we consider the 3D Schr¨odinger operators with constant magnetic field of strength b > , pointing at the x -direction. For the magnetic potential A = ( − b x , b x , it is given by : H := − (cid:16) ∇ A (cid:17) = (cid:16) D + b x (cid:17) + (cid:16) D − b x (cid:17) + D , D j := − i ∂∂x j , ∇ Aj := ∇ x j − iA j . (1.1) Mathematics Subject Classification.
Key words and phrases.
Magnetic Schr¨odinger operator, Boundary conditions, Counting function of resonances.
Set x ⊥ := ( x , x ) ∈ R . Using the representation L ( R ) = L ( R x ⊥ ) ⊗ L ( R x ) , H admits thedecomposition(1.2) H = H Landau ⊗ I + I ⊥ ⊗ (cid:16) − ∂ ∂x (cid:17) with(1.3) H Landau := (cid:16) D + b x (cid:17) + (cid:16) D − b x (cid:17) , the Landau Hamiltonian and I and I ⊥ being the identity operators in L ( R x ) and L ( R x ⊥ ) respec-tively. The spectrum of H Landau consists of the so-called Landau levels Λ q = (2 q + 1) b , q ∈ N := { , , , . . . } , and dim Ker( H Landau − Λ q ) = ∞ . Consequently, σ ( H ) = σ ac ( H ) = [ b, + ∞ [ , and the Landau levels play the role of thresholds in the spectrum of H .Let us introduce the obstacle perturbation. Let K ⊂ R be a compact domain with smooth bound-ary Σ and let Ω := R \ K . We denote by ν the unit outward normal vector of the boundary Σ and by ∂ AN := ∇ A · ν the magnetic normal derivative. For γ a smooth real valued function on Γ , we introducethe following operator on Σ : ∂ A,γ Σ := ∇ A · ν + γ. From now, γ is fixed and if it does not lead to confusion, we shall omit the index A, γ and write ∂ Σ for ∂ A,γ Σ .In the following lines let us define H γ Ω (resp. H ∞ Ω ) the Neumann and Robin (resp. Dirichlet)realization of − (cid:16) ∇ A (cid:17) on Ω . Neumann and Robin realizations of − (cid:16) ∇ A (cid:17) : The operator H γ Ω is defined by(1.4) H γ Ω u = − (cid:16) ∇ A (cid:17) u, u ∈ Dom (cid:0) H γ Ω (cid:1) ,Dom (cid:0) H γ Ω (cid:1) := n u ∈ L (Ω) : (cid:16) ∇ A (cid:17) j u ∈ L (Ω) , j = 1 , ∂ A,γ Σ u = 0 on Σ o . Actually, H γ Ω is the self-adjoint operator associated to the closure of the quadratic form(1.5) Q γ Ω ( u ) = Z Ω (cid:12)(cid:12) ∇ A u (cid:12)(cid:12) dx + Z Σ γ | u | dσ, x := ( x ⊥ , x ) , originally defined in the magnetic Sobolev space H A (Ω) := (cid:8) u ∈ L (Ω) : ∇ A u ∈ L (Ω) (cid:9) .The Neumann realization corresponds to γ = 0 . Dirichlet realization of − (cid:16) ∇ A (cid:17) : The operator H ∞ Ω is defined by(1.6) H ∞ Ω u = − (cid:16) ∇ A (cid:17) u, u ∈ Dom (cid:0) H ∞ Ω (cid:1) ,Dom (cid:0) H ∞ Ω (cid:1) := n u ∈ L (Ω) : (cid:16) ∇ A (cid:17) j u ∈ L (Ω) , j = 1 , u = 0 on Σ o . AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 3
Actually, H ∞ Ω is the self-adjoint operator associated to the closure of the quadratic form(1.7) Q ∞ Ω ( u ) = Z Ω (cid:12)(cid:12) ∇ A u (cid:12)(cid:12) dx, x := ( x ⊥ , x ) , originally defined on C ∞ (Ω) . Remark 1.1.
The magnetic Schr¨odinger operator H defined by (1.1) is the self-adjoint operatorassociated to the closure of the quadratic form (1.7) with Ω = R . As compactly supported perturbations of the elliptic operator H , the operators H ∞ Ω and H γ Ω arerelatively compact perturbations of H and we have: Proposition 1.2.
For l = ∞ , γ , the essential spectrum of H l Ω coincide with these of H : σ ess ( H ∞ Ω ) = σ ess ( H γ Ω ) = σ ess ( H ) = σ ( H ) = [ b, + ∞ ) . This result is proved in a more general context in [13]. It is also a consequence of some resolventequations as in Section 3.In order to define the resonances, let us recall analytic properties of the free resolvent. Let M bethe connected infinite-sheeted covering of C \ ∪ q ∈ N { Λ q } where each function z p z − Λ q , q ∈ N is analytic. Near a Landau level Λ q this Riemann surface M can be parametrized by z q ( k ) = Λ q + k , k ∈ C ∗ , | k | ≪ (for more details, see Section 2 of [3]). For ǫ > , we denote by M ǫ the set of thepoints z ∈ M such that for each q ∈ N , we have Im p z − Λ q > − ǫ . We have ∪ ǫ> M ǫ = M . Proposition 1.3. [3, Proposition 1]
For each ǫ > , the operator R ( z ) = ( H − z ) − : e − ǫ h x i L ( R ) → e ǫ h x i L ( R ) has a holomorphic extension (still denoted by R ( z ) ) from the open upper half-plane C + := { z ∈ C ; Im z > } to M ǫ . Since H ∞ Ω and H γ Ω are compactly supported perturbations of H , using some resolvent equationsand the analytic Fredholm theorem, from Proposition 1.3, we deduce meromorphic extension of theresolvents of H ∞ Ω and H γ Ω . It can be done following the ”black box” framework developed for per-turbation of the Laplacian (as in [22], [20]) or by introducing auxiliary operators as in Section 3 (seeCorollary 3.4). Then we are able to define the resonances: Definition 1.4.
For l = ∞ , γ , we define the resonances for H l Ω as the poles of the meromorphicextension of the resolvent ( H l Ω − z ) − : e − ǫ h x i L (Ω) → e ǫ h x i L (Ω) . These poles (i.e. the resonances) and the rank of their residues (the multiplicity of the resonance) donot depend on ǫ > . Our goal is to study the distribution of the resonances of H ∞ Ω and H γ Ω near the Landau levels. Wewill essentially prove that the distribution of the resonances of H ∞ Ω (resp. H γ Ω ) near the Landau levelsis essentially governed by the distribution of resonances of H + K (resp. H − K ) which is knownthanks to [4] .The article is organized as follows. Our main results and their corollaries are formulated and dis-cussed in Section 2. In Section 3, we show how we can reduce the study of the operators H ∞ Ω and H γ Ω V. BRUNEAU AND D. SAMBOU near the Landau levels to some compact perturbations, of fixed sign, of H − . By this way, in Subsec-tion 3.3, we bring out the relation between the perturbed operators of H − and the Dirichlet-Neumannand the Neumann-Dirichlet operators. Section 4 is devoted to the proofs of our main results. In Sec-tions 5 and 6, exploiting the fact that the Dirichlet-Neumann and the Neumann-Dirichlet operators areelliptic pseudo differential operators on the boundary Σ , we show how we can reduce the analyse ofthe perturbed operators to that of Toeplitz operators with symbol supported near the obstacle. Section7 is devoted to the computational proof of the lemma needed to prove that the Diric hlet-Neumannand the Neumann-Dirichlet operators are elliptic pseudo differential operators on the boundary Σ .2. F ORMULATION OF THE MAIN RESULTS
For l = ∞ , γ , let H l Ω be the magnetic Schr¨odinger operators defined by (1.4) and (1.6) and letus denote by Res (cid:0) H l Ω (cid:1) the corresponding resonances sets.Near a Landau level Λ q , q ∈ N , we parametrize the resonances z q by z q ( k ) = Λ q + k with | k | << .Our main result gives the localization of the resonances of H ∞ Ω and H γ Ω near the Landau levels Λ q , q ∈ N , together with an asymptotic expansion of the resonances counting function in small annulusadjoining Λ q , q ∈ N . As consequences we obtain some informations concerning eigenvalues. Theorem 2.1.
Let K ⊂ R be a smooth compact domain. Fix a Landau level Λ q , q ∈ N , suchthat K does not produce an isolated resonance at Λ q (see Definition 4.7). Then the resonances z q ( k ) = Λ q + k of H ∞ Ω and H γ Ω , with | k | << sufficiently small, satisfy: (i) For < r < √ b fixed and l = ∞ , γ X z q ( k ) ∈ Res ( H l Ω ) r< | k |
The Robin (resp. Neumann) exterior operator H γ Ω (resp. H ) has an increasingsequence of eigenvalues { µ j } j which accumulate at Λ with the distribution: { µ j ∈ σ p ( H γ Ω ) ∩ ( −∞ , Λ − λ ) } ∼ | ln λ | | ln λ | (cid:0) o (1) (cid:1) , λ ց , (ii) The Dirichlet exterior operator H ∞ Ω has no eigenvalues below Λ . AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 5
Moreover, since the embedded eigenvalues of the operator H l Ω in [ b, + ∞ ) \ ∪ ∞ q =0 { Λ q } are theresonances z q ( k ) with k ∈ e i { , π } ]0 , √ b [ , then an immediate consequence of Theorem 2.1 (ii) and (iii) is the absence of embedded eigenvalues of H ∞ Ω in ]Λ q − r , Λ q [ ∪ ]Λ q , Λ q + r [ and of embeddedeigenvalues of H γ Ω in ]Λ q , Λ q + r [ , for r sufficiently small. Hence we have the following result: Corollary 2.3. In [ b, + ∞ ) \ ∪ ∞ q =0 (cid:8) Λ q (cid:9) (resp. in [ b, + ∞ ) \ ∪ ∞ q =1 (cid:8) ]Λ q − r , Λ q [ (cid:9) ) the embeddedeigenvalues of the operator H ∞ Ω (resp. H γ Ω ), form a discrete set. Re( k )Im( k ) r r S θ θ θ × ×× ××× × × ××××××××××××××××××××× ×××××××××××× Dirichlet
Re( k )Im( k ) r r θ × × × ×××××××××××××× ×× ××××××× Neumann-Robin S θ F IGURE Localisation of the resonances in variable k : For r sufficiently small,the resonances z q ( k ) = Λ q + k of the operators H l Ω , l = ∞ , γ near a Landau level Λ q , q ∈ N , are concentrated in the sectors S θ . For l = ∞ they are concentrated nearthe semi-axis − i ]0 , + ∞ ) in both sides, while they are concentrated near the semi-axis i ]0 , + ∞ ) on the left for l = γ .To our best knowledge, the above results are new even concerning the discret spectrum. However,they are not surprising. Similar results hold for perturbations by potentials (see [18] for eigenvalues, et[3], [4] for resonances) and for exterior problems in the 2D case concerning accumulation of eigenval-ues at the Landau levels (see [17], [16], [10]). In comparison with previous works, the spectral studyof obstacle perturbations in the 3D case leads to two new difficulties. The first, with respect to the 2Dcase, comes from the presence of continuous spectrum, then the spectral study involves resonancesand some non-selfadjoint aspects. The second difficulty, with respect to the potential perturbations,is due to the fact that the perturbed and the unperturbed operators are not defined on the same space.In order to overcome this difficulty, we introduce an appropriate perturbation V l , l = ∞ , γ , of H − on L ( R ) in such a way that the concentration of resonances of H l Ω , l = ∞ , γ at Λ q is reduced tothe accumulation of ” Birman-Schwinger singularities ” of H − − V l at q in the sense that z is a” Birman-Schwinger singularity ” of H − − V l if is an eigenvalue of B l ( z ) := sign ( V l ) | V l | (cid:16) H − − z (cid:17) − | V l | = sign ( V l ) | V l | zH ( H − z ) − | V l | = zV l + z sign ( V l ) | V l | ( H − z ) − | V l | (see Section 3 and in particular Proposition 3.3). Then the main tool of our proof is an abstract resultof [4] (see Section 4 and especially Proposition 4.2). V. BRUNEAU AND D. SAMBOU
3. M
AGNETIC RESONANCES FOR THE EXTERIOR PROBLEMS
In this section we reduce the study of the operators H ∞ Ω and H γ Ω near the Landau levels to somecompact perturbations, of fixed sign, of H − . We follow ideas developped in [17] and [10] for theeigenvalues of the D Schr¨odinger operators and give a charaterisation of the resonances which willallow to apply (in Section 4) a general result of [4].3.1.
Auxiliary operators.
By identification of L ( R ) with L (Ω) ⊕ L ( K ) , we consider thefollowing operator in L ( R ) :(3.1) ˜ H γ := H γ Ω ⊕ H − γK on Dom ( H γ Ω ) ⊕ Dom ( H − γK ) , where H − γK is the Robin operator in K . Namely, H − γK is the self-adjoint operator associated to theclosure of the quadratic form Q − γK defined by (1.5), by replacing γ and Ω with − γ and K respectively.Without loss of generality, we can assume that H γ Ω , H − γK and H ∞ Ω are positive and invertible (if notit is sufficient to shift their by the same constant), and we introduce(3.2) V γ := H − − ( ˜ H γ ) − = H − − ( H γ Ω ) − ⊕ ( H − γK ) − . (3.3) V ∞ := H − − ( H ∞ Ω ) − ⊕ . On one hand, thanks to the choice of the boundary condition in K (with − γ ), the quadratic formassociated to ˜ H γ is given by(3.4) ˜ Q γ ( u Ω , u K ) = Q γ Ω ( u Ω ) + Q − γK ( u K ) = Z R (cid:12)(cid:12) ∇ A u (cid:12)(cid:12) dx. Thus, d ( ˜ Q γ ) , the domain of the quadratic form associated to ˜ H γ contains H A ( R ) the domain of Q ,the quadratic form associated to H : Q ( u ) = Z R (cid:12)(cid:12) ∇ A u (cid:12)(cid:12) dx, and ˜ Q γ coincide with Q on H A ( R ) .On the other hand by extending by the functions of the domain of the quadratic form Q ∞ Ω (see(1.7)) we can embed d ( Q ∞ Ω ) in H A ( R ) with Q ∞ Ω coinciding with Q on d ( Q ∞ Ω ) .From the previous properties, according to Proposition 2.1 of [17], we deduce that V γ (defined by(3.2)) is a non positive operator and V ∞ (defined by (3.3)) is non negative in L ( L ( R )) .3.2. Decreasing and compact properties of the perturbations of fixed sign.Lemma 3.1.
The operators V ∞ and V γ defined by (3.3) and (3.2) are respectively non negativeand non positive compact operators in L ( L ( R )) . Moreover, there exists M ∞ and M γ compactoperators in L ( L ( R )) such that (3.5) V ∞ = M ∞ M ∞ , V γ = − M γ M γ ; M l := ( M l ) ∗ , l = γ, ∞ , with M l , l = γ, ∞ bounded from e ǫ h x i L ( R ) into L ( R ) , < ǫ < √ b . AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 7
Proof.
Since V ∞ and ( − V γ ) are non negative bounded operators on L ( R ) (see Subsection 3.1above) there exists bounded operators M l , l = γ, ∞ such that V ∞ = ( M ∞ ) ∗ M ∞ , and V γ = − ( M γ ) ∗ M γ . For example, we can take M l = ( M l ) ∗ = | V l | , but sometimes other choices could bemore convenient (see remark 3.2).For all f, g ∈ L ( R ) , let us introduce v = H − f , u Ω ,l = ( H l Ω ) − ( g | Ω ) and u K = ( H − γK ) − ( g | K ) .By definition of V l , l = γ, ∞ , we have: h f, V ∞ g i L ( R ) = h H v, (cid:0) H − − ( H ∞ Ω ) − ⊕ (cid:1)(cid:0) H ∞ Ω u Ω , ∞ ⊕ g | K (cid:1) i = h v, H ∞ Ω u Ω , ∞ ⊕ g | K i − h H v, u Ω , ∞ ⊕ i = − Z Ω v ( ∇ A ) u Ω , ∞ dx + Z Ω ( ∇ A ) v u Ω , ∞ dx + Z K v g | K dx, and h f, V γ g i L ( R ) = h H v, (cid:0) H − − ( ˜ H γ ) − (cid:1)(cid:0) H γ Ω u Ω ,γ ⊕ H − γK u K (cid:1) i = h v, (cid:0) H γ Ω u Ω ,γ ⊕ H − γK u K (cid:1) i − h H v, u Ω ,γ ⊕ u K i = − Z Ω v ( ∇ A ) u Ω ,γ dx + Z Ω ( ∇ A ) v u Ω ,γ dx − Z K v ( ∇ A ) u K dx + Z K ( ∇ A ) v u K dx. Then by integration by parts, from the boundary conditions u Ω , ∞| Σ = 0 and ∂ Σ u K = 0 = ∂ Σ u Ω ,γ ,we deduce the equalities(3.6) h f, V ∞ g i L ( R ) = Z K v | K g | K dx + Z Σ Γ ( v ) ∂ AN u Ω , ∞ dσ, (3.7) h f, V γ g i L ( R ) = − Z Σ ∂ Σ v Γ (cid:0) u Ω ,γ (cid:1) − Γ (cid:0) u K (cid:1) dσ, where Γ : H s ( • ) −→ H s − (Σ) , for • = Ω , K , and s ≥ , is the trace operator on Σ . In the notationof this operator, we omit the dependence on K or Ω because either it is indicate on the functions onwhich it is applied, or the functions are smooth near Σ . In particular, due to the regularity propertiesof v = H − f near Σ , the functions Γ ( v ) and ∂ Σ v are well defined.In other words, we have(3.8) h f, V ∞ g i L ( R ) = h ( H − f ) | K , g | K i L ( K ) + h Γ ( H − f ) , ∂ AN ( H ∞ Ω ) − ( g | Ω ) i L (Σ) , (3.9) h f, V γ g i L ( R ) = −h ∂ Σ ( H − f ) , Γ ( H γ Ω ) − ( g | Ω ) − Γ ( H − γK ) − ( g | K ) i L (Σ) . Exploiting that the domains of the operators contain H loc and the compacity of the domains K and Σ ,we deduce that V ∞ and V γ are compact operators in L ( R ) .At last, in order to prove that M ∞ and M γ are bounded from e ǫ h x i L ( R ) into L ( R ) , let usprove that for l = γ, ∞ , e ǫ h x i V l e ǫ h x i is bounded in L ( R ) .Clearly, in the relations (3.6) and (3.7), v can be replaced by χ v for any χ ∈ C ∞ c ( R x ) equals to on(3.10) I K := ∪ ( x ,x ) ∈ R { x ; ( x , x , x ) ∈ K } , V. BRUNEAU AND D. SAMBOU ( χ = 1 near K is sufficient) and consequently, for l = γ, ∞ , we have h f, V l g i = h f χ , V l g i with f χ = H ( χ v ) = ( H χ H − ) f . Thus, V l = H − χ H V l and taking the adjoint relation wededuce: V l = V l H χ H − = H − χ H V l H χ H − By using the orthogonal decomposition of H − : H − = X q ∈ N p q ⊗ ( D x + Λ q ) − with ( D x + Λ q ) − having the integral kernel e − √ Λ q | x − x ′ | √ Λ q , we deduce that H χ H − = χ +[ D x , χ ] H − is bounded from e ǫ h x i L ( R ) into L ( R ) for < ǫ < √ b and then so is M l , l = γ, ∞ . (cid:3) Remark 3.2.
As written in the above proof, we can take M l = ( M l ) ∗ = | V l | , but sometimes otherchoices could be more convenient. In particular in order to reduce our analyse to the boundary Σ , itcould be interesting to consider operator M l from L ( R ) into L (Σ) by exploiting the link with theDirichlet-Neumann and Robin-Dirichlet operators (see Subsection 3.3). Relation with Dirichlet-Neumann and Robin-Dirichlet operators.
Taking g = f = H v and by introducing w Ω , ∞ := v | Ω − u Ω , ∞ in (3.6), we obtain:(3.11) h H v, V ∞ H v i L ( R ) = Z K v | K f | K dx + Z Σ Γ ( v ) ∂ AN v | Ω dσ − Z Σ Γ ( v ) ∂ AN w Ω , ∞ dσ, with w Ω , ∞ satisfying(3.12) (cid:26) ( ∇ A ) w Ω , ∞ = 0 in ΩΓ ( w Ω , ∞ ) = Γ ( v ) . On the same way, from (3.7), we have(3.13) h H v, V γ H v i L ( R ) = − Z Σ ∂ Σ v Γ (cid:0) w K,γ (cid:1) − Γ (cid:0) w Ω ,γ (cid:1) dσ, with w Ω ,γ = v | Ω − u Ω ,γ and w K,γ = v | K − u K satisfying for • = Ω , K (3.14) (cid:26) ( ∇ A ) w • ,γ = 0 in • ∂ Σ w • ,γ = ∂ Σ v. Consequently, V ∞ and V γ are related to the Dirichlet-Neumann and
Robin-Dirichlet operators:(3.15) h H v, V ∞ H v i L ( R ) = h v | K , ( H v ) | K i L ( K ) + h Γ ( v | Ω ) , ∂ AN ( v | Ω ) − DN Ω (Γ v | Ω ) i L (Σ) , (3.16) h H v, V γ H v i L ( R ) = −h ∂ Σ v , ( RD K − RD Ω ) ∂ Σ v i L (Σ) . For the definition and elliptic properties of these operators (on some subspaces of finite codimension),we refer to Proposition 6.4.
AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 9
Definition and characterisation of the resonances.
Let us introduce, for
Im( z ) > , thebounded operators(3.17) ˜ R γ ( z ) = ( ˜ H γ − z ) − = ( H γ Ω − z ) − ⊕ ( H − γK − z ) − and ˜ R ∞ ( z ) = ( H ∞ Ω − z ) − ⊕ . Proposition 3.3.
For l = γ, ∞ , the operator-valued function ˜ R l ( z ) : e − ǫ h x i L ( R ) −→ e ǫ h x i L ( R ) has a meromorphic extension (cid:0) also denoted ˜ R l ( · ) (cid:1) from the open upper half plane to M ǫ , ǫ < √ b .Moreover, the following assertions are equivalent: a) z is a pole of ˜ R l in L (cid:16) e − ǫ h x i L ( R ) , e ǫ h x i L ( R ) (cid:17) , b) z is a pole of M l ˜ R l M l in L (cid:16) L ( R ) (cid:17) , c) − is an eigenvalue of ε ( l ) B l ( z ) with (3.18) B l ( z ) := zM l H R ( z ) M l = zM l M l + z M l R ( z ) M l , where ε ( ∞ ) = 1 , ε ( γ ) = − .Proof. For
Im( z ) > and l = γ, ∞ we have the resolvent equation: ˜ R l ( z ) (cid:16) I + zV l H ( H − z ) − (cid:17) = ˜ R l (0) H ( H − z ) − = ( H − z ) − − V l H ( H − z ) − . Let us denote by e ± the multiplication operator by e ± ǫ h x i . Then, by introducing e ± , and writing H ( H − z ) − = I + z ( H − z ) − , we have e − ˜ R l ( z ) e − (cid:16) I + ze + V l e − + z e + V l ( H − z ) − e − (cid:17) = e − ( H − z ) − e − − e − V l e − − ze − V l ( H − z ) − e − . Since e − ( H − z ) − e − admits a holomorphic extension from the open upper half plane to M ǫ (cid:0) seeProposition 1 of [3] (cid:1) , according to Lemma 3.1, e + V l ( H − z ) − e − and e − V l ( H − z ) − e − can beholomorphically extended to M ǫ . Then from the Fredholm analytic Theorem we deduce the mero-morphic extension of z ˜ R l ( z ) from the open upper half plane to M ǫ .Moreover, by writing M l ˜ R l ( z ) M l = ( M l e + ) e − ˜ R l ( z ) e − ( e + M l ) , we show the holomorphic extension of M l ˜ R l M l in L (cid:16) L ( R ) (cid:17) with poles among those of e − ˜ R l e − .Conversely, according to the following resolvent equation, the poles of e − ˜ R l e − are those of M l ˜ R l M l in L (cid:16) L ( R ) (cid:17) : ˜ R l ( z ) = ( H − z ) − − V l H ( H − z ) − − zH ( H − z ) − ˜ R l (0) V l H ( H − z ) − + z H ( H − z ) − V l ˜ R l ( z ) V l H ( H − z ) − . We conclude the proposition from the equation:(3.19) (cid:16) I + ε ( l ) M l (cid:16) z − H − (cid:17) − M l (cid:17) (cid:16) I − ε ( l ) M l (cid:16) z − ˜ R l (0) (cid:17) − M l (cid:17) = I, and using that M l (cid:16) z − H − (cid:17) − M l = zM l H ( H − z ) − M l = zM l M l + z M l ( H − z ) − M l , M l (cid:16) z − ˜ R l (0) (cid:17) − M l = zM l M l + z M l ˜ R l ( z ) M l . (cid:3) By definition of ˜ R l , l = ∞ , γ , (see (3.17)), we also have: Corollary 3.4.
For l = γ, ∞ , the operator-valued function ( H l Ω − z ) − : e − ǫ h x i L (Ω) −→ e ǫ h x i L (Ω) has a meromorphic extension, denoted R l Ω , from the open upper half plane to M ǫ , ǫ < √ b .Moreover, according to their multiplicities (i.e. the rank of their residues), the poles of R ∞ Ω coincidewith the poles of ˜ R ∞ and the poles of R γ Ω are those of ˜ R γ excepted the eigenvalues of H − γK .
4. O
UTLINE OF PROOFS
In order to prove our main results, in this section, let us begin by recalling some auxiliary resultsconcerning characteristic values of holomorphic operators due to Bony, the first author and Raikov[4]. Then we will apply these auxiliary results to our problem and prove the main results.4.1.
Auxiliary results.
Let D be a domain of C containing zero and let us consider an holomorphicoperator-valued function A : D −→ S ∞ , where S ∞ is the class of compact operators in a separableHilbert space. Definition 4.1.
For a domain ∆ ⊂ D \ { } , a complex number z ∈ ∆ is a characteristic value of z I − A ( z ) z if the operator I − A ( z ) z is not invertible. The multiplicity of a characteristic value z is defined by (4.1) mult ( z ) := 12 iπ tr (cid:16) Z C (cid:0) − A ( z ) z (cid:1) ′ (cid:0) I − A ( z ) z (cid:1) − dz (cid:17) , where C is a small contour positively oriented containing z as the unique point z satisfying (cid:0) I − A ( z ) z (cid:1) is not invertible. Let us denote by Z (∆ , A ) , the set of the characteristic values of (cid:0) I − A ( z ) z (cid:1) inside ∆ : Z (∆ , A ) := (cid:26) z ∈ ∆ : I − A ( z ) z is not invertible (cid:27) . If there exists z ∈ ∆ such that I − A ( z ) z is not invertible, then Z (∆ , A ) is a discrete set (see e.g. [11,Proposition 4.1.4]).Assume that A (0) is self-adjoint and for T a compact self-adjoint operator, let us introduce thecounting function(4.2) n ( r, T ) := Tr [ r, + ∞ ) ( T ) , the number of eigenvalues of the operator T lying in the interval [ r, + ∞ ) ⊂ R ∗ , and counted withtheir multiplicity. Denote by Π the orthogonal projection onto ker A (0) .As consequence of [4, Corollary 3.4, Theorem 3.7, Corollary 3.11], we have the following resultwhich states that the characteristic values of z I − A ( z ) z are localized near the real axis where A (0) has its spectrum, and the distribution of the characteristic values near is governed by the distributionof the eigenvalues of A (0) near . AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 11
Proposition 4.2. [4]
Let A be as above and I − A ′ (0)Π be invertible. Assume that ∆ ⋐ C \ { } is a bounded domainwith smooth boundary ∂ ∆ which is transverse to the real axis at each point of ∂ ∆ ∩ R . We have: (i) The characteristic values z ∈ Z (∆ , A ) near satisfy | Im( z ) | = o ( | z | ) . (ii) If the operator A (0) has a definite sign ( ± A (0) ≥ , then the characteristic values z near satisfy ± Re( z ) ≥ . (iii) For ± A (0) ≥ , if the counting function of A (0) satisfies: n ( r, ± A (0)) = c | ln r | ln | ln r | (1 + o (1)) , r ց , then, for r > fixed, the counting function of the characteristic values near satisfies: (cid:8) z ∈ Z (∆ , A ); r < | z | < r (cid:9) = c | ln r | ln | ln r | (1 + o (1)) , r ց . Preliminary results.
In this subsection we apply the previous abstract results to our problem.
Proposition 4.3.
Fix q ∈ N . Then z q ( k ) = Λ q + k , < | k | ≪ isapole of ˜ R l (cid:0) defined by (3.17) (cid:1) in L ( e − ǫ h x i L , e ǫ h x i L ) ifand only if(4.3) I − ε ( l ) A lq ( ik ) ik is not invertible , ε ( l ) := ( if l = ∞− if l = γ , where z A lq ( z ) ∈ S ∞ ( L ( R )) isthe holomorphic operator-valued function given by(4.4) A lq ( ik ) = z q ( k ) M l ( p q ⊗ r ( ik )) M l − ikM l (cid:16) z q ( k ) + z q ( k ) R ( z q ( k ))( I − p q ⊗ I ) (cid:17) M l , with r ( z ) theintegral operator in L ( R x ) whose integral kernel is e z | x − x ′ | , x , x ′ ∈ R .In particular, A lq (0) = Λ q M l ( p q ⊗ r (0)) M l is a non negative compact operator whose countingfunction satisfies(4.5) n (cid:0) r, A lq (0) (cid:1) = n (cid:0) r, T lq (cid:1) ; T lq := ε ( l )Λ q p q W l p q with W l defined on L ( R ) by(4.6) ( W l f ⊥ )( x ⊥ ) = 12 Z R x (cid:16) V l ( f ⊥ ⊗ R ) (cid:17) ( x ⊥ , x ) dx , Proof.
From Proposition 3.3, z q ( k ) = Λ q + k is a pole of ˜ R l ( . ) if and only if I + ε ( l ) B l (Λ q + k ) is not invertible with B l (Λ q + k ) := z q ( k ) M l M l + z q ( k ) M l R (Λ q + k ) M l . We split the sandwiched resolvent M l R ( z ) M l into two parts as follows(4.7) M l R ( z ) M l = M l R ( z )( p q ⊗ I ) M l + M l R ( z )( I − p q ⊗ I ) M l . For z q ( k ) = Λ q + k in the resolvent set of the operator H , we have ( H − Λ q − k ) − = X j ∈ N p j ⊗ (cid:0) D x + Λ j − Λ q − k (cid:1) − . Hence by definition of p q , M l R (Λ q + k )( I − p q ⊗ I ) M l is holomorphic near k = 0 (for moredetails, see the proof of Proposition 1.3). Furthermore, for k chosen such that Im( k ) > , we have(4.8) M l R ( z q ( k ))( p q ⊗ I ) M l = M l p q ⊗ (cid:0) D x − k (cid:1) − M l = − M l ( p q ⊗ r ( ik )) M l ik , where r ( z ) is the integral operator introduced above. Hence (4.3) and (4.4) hold because B l (Λ q + k ) = − A lq ( ik ) ik . Let us compute the operator A lq (0) for l = γ, ∞ . We have(4.9) A lq (0) = Λ q M l ( p q ⊗ r (0)) M l , l = γ, ∞ , where r (0) is the operator acting from e − ǫ h x i L ( R ) into e ǫ h x i L ( R ) with integral kernel given bythe constant function .Now from Lemma 3.1, it follows that there exists a bounded operator M l on L ( R ) such that M l = M l e − ǫ h x i for l = γ, ∞ . Recalling that e ± is the multiplication operator by e ± := e ± ǫ h x i , itcan be easily checked that(4.10) M l ( p q ⊗ r (0)) M l = M l e + ( p q ⊗ c ∗ c ) e + M l B ∗ q,l B q,l , where c : L ( R ) −→ C is the operator defined by c ( u ) := h u, e − i , so that c ∗ : C −→ L ( R ) is givenby c ∗ ( λ ) = λe − , and(4.11) B q,l := 1 √ p q ⊗ c ) e + M l , l = γ, ∞ . More explicitly, the operator B q,l satisfies B q,l : L ( R ) −→ L ( R ) with ( B q,l ϕ )( x ⊥ ) = 1 √ Z R P q,b ( x ⊥ , x ′⊥ )( M l ϕ )( x ′⊥ , x ′ ) dx ′⊥ dx ′ , where P q,b ( · , · ) is the integral kernel of p q given by:(4.12) P q,b ( x ⊥ , x ′⊥ ) = b π L q (cid:18) b | x ⊥ − x ′⊥ | (cid:19) exp (cid:16) − b (cid:0) | x ⊥ − x ′⊥ | + 2 i ( x x ′ − x ′ x ) (cid:1)(cid:17) , with x ⊥ = ( x , x ) , x ′⊥ = ( x ′ , x ′ ) ∈ R ; here L q ( t ) := q ! e t d q ( t q e − t ) dt q are the Laguerre polynomials.The adjoint operator B ∗ q,l : L ( R ) −→ L ( R ) satisfies ( B ∗ q,l f ⊥ )( x ⊥ , x ) = 1 √ M l ( p q f ⊥ ⊗ R )( x ⊥ , x ) , where ( p q f ⊥ ⊗ R )( x ⊥ , x ) = p q f ⊥ ( x ⊥ ) is constant with respect to x . Thus, from (4.9) and (4.10), A lq (0) = Λ q B ∗ q,l B q,l is a positive compact self-adjoint operator with the non zero eigenvalues equalto those of Λ q B q,l B ∗ q,l : L ( R ) −→ L ( R ) given by(4.13) Λ q B q,l B ∗ q,l = ε ( l )Λ q p q W l p q = T lq , l = γ, ∞ , where W l is defined on L ( R ) by (4.6). Here, we have used that M l M l = ε ( l ) V l (see Lemma 3.1).Then (4.14) n (cid:0) r, A lq (0) (cid:1) = n (cid:0) r, Λ q B ∗ q,l B q,l (cid:1) = n (cid:0) r, Λ q B q,l B ∗ q,l (cid:1) = n (cid:0) r, T lq (cid:1) . AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 13 (cid:3)
Remark 4.4.
Here, we omit the proof that the multiplicity of the pole z q ( k ) of ˜ R l (i.e. the rank ofthe residue) coincides with the multiplicity of k as a characteristic value of k I − ǫ ( l ) A lq ( ik ) ik (cid:0) see (4.1) (cid:1) . It is closely related to the proof of Proposition 3 of [3] . In order to analyse the counting function n (cid:0) r, A lq (0) (cid:1) = n (cid:0) r, T lq (cid:1) for l = γ, ∞ , we will need thefollowing result. Proposition 4.5.
For all q ∈ N , there exists L q a finite codimension subspaces of ker ( H Landau − Λ q ) and there exists C > , such that for compacts domains K ⊥ ⊂ K ⊥ ⊂ R and compact intervals I ⊂ I satisfying K ⊥ × I ⊂ K ⊂ K ⊥ × I with ∂ ( K ⊥ i × I i ) ∩ ∂K = ∅ , i = 0 , , we have for any f ⊥ ∈ L q , (4.15) C h f ⊥ , p q K ⊥ p q f ⊥ i L ( R ) ≤ h f ⊥ , T lq f ⊥ i L ( R ) ≤ C h f ⊥ , p q K ⊥ p q f ⊥ i L ( R ) , where T lq is defined by (4.5) , l = ∞ , γ . The proof of Proposition 4.5 will be given in Section 6 by introducing elliptic pseudo-differentialoperators on the boundary
Σ = ∂ Ω = ∂K .4.3. Proof of Theorem 2.1.
First, near Λ q , Corollary 3.4 allows to reduce the study of the reso-nances of H l Ω , z q ( k ) = Λ q + k , | k | ≤ r ( r sufficiently small) to the poles of ˜ R l , because theycoincide for l = ∞ and for l = γ there is only a finite number of eigenvalues of H − γK near Λ q .Then, thanks to Proposition 4.3, modulo a finite set F q , the resonances near Λ q are related to thecharacteristic values of z I − A lq ( ε ( l ) z ) z :(4.16) { z q ( k ) = Λ q + k ∈ Res ( H l Ω ) ∩ B (0 , r ) ∗ } = (cid:8) z q ( k ) = Λ q + k ; such that ε ( l ) ik ∈ Z (cid:0) B (0 , r ) ∗ , A lq ( ε ( l ) · ) (cid:1)(cid:9) \ F q , where B (0 , r ) ∗ = { k ∈ C ; 0 < | k | ≤ r } and A lq , l = ∞ , γ , is defined by (4.4).Then, provided that(4.17) I − ε ( l )( A lq ) ′ (0)Π q is an invertible operator , (cid:0) with Π q the orthogonal projection onto ker A lq (0) (cid:1) , (ii) and (iii) of Theorem 2.1 are immediate con-sequences of (i) and (ii) of Proposition 4.2 with z = ε ( l ) ik , because A lq (0) is non negative.Let us prove (i) of Theorem 2.1 and interpret (4.17). In order to apply (iii) of Proposition 4.2,we analyse the counting function of the eigenvalues of A lq (0) . According to (4.5), n (cid:0) r, A lq (0) (cid:1) = n (cid:0) r, T lq (cid:1) , T lq = ε ( l )Λ q p q W l p q . This together with Proposition 4.5, by using the mini-max principle,implies that for l = ∞ , γ ,(4.18) n (cid:0) Cr, p q K ⊥ p q (cid:1) ≤ n (cid:0) r, A lq (0) (cid:1) ≤ n (cid:0) r/C, p q K ⊥ p q (cid:1) . Since K i for i = 0 , are compact sets (with nonempty interior), then according to [19, Lemma 3.5]we have n (cid:0) r, p q K ⊥ i p q (cid:1) = | ln r | ln | ln r | (cid:0) o (1) (cid:1) as r ց . Combining this with (4.18), we deduce(4.19) n (cid:0) r, A lq (0) (cid:1) = | ln r | ln | ln r | (cid:0) o (1) (cid:1) , r ց . We conclude the proof of (i) of Theorem 2.1 from (4.16), (iii) of Proposition 4.2, (4.19) and the fol-lowing proposition giving the interpretation of the technical assumption (4.17) in terms of resonances.4.4.
Interpretation of the assumption (4.17) . Using the notations of Subsection 4.2, let us intro-duce the operator(4.20) P q := p q ⊗ r (0) : e − ǫ h x i L ( R ) −→ e ǫ h x i L ( R ) , ǫ > , whose integral kernel is P q,b ( x ⊥ , x ′⊥ ) (see (4.12)). Proposition 4.6.
Let A lq be the holomorphic operator-valued function defined by (4.4) and Π q theorthogonal projection onto Ker A lq (0) . Then the following assertions are equivalent: (i) I − ε ( l )( A lq ) ′ (0)Π q is invertible (ii) The following limit exists for z in a sector S δ := (cid:8) z ∈ C ; Im( z ) > δ | Re( z ) − Λ q | (cid:9) , δ > : lim S δ ∋ z −→ Λ q M l ( I + ikP q ) ˜ R l ( z ) M l , where k = p z − Λ q , Im( k ) > , Re( k ) > , and ˜ R l ( z ) is defined by (3.17) . Let us recall that (i) is the assumption (4.17), and before to prove the above result let us give aninterpretation of (ii) . From (4.7) and (4.8), z = Λ q is an essential singularity of M l ( H − z ) − M l given by M l ( H − z ) − M l = − ik M l P q M l + Hol Λ q ( z ) , where Hol Λ q is a holomorphic operator valued function near z = Λ q , given byHol Λ q ( z ) = M l ( I + ikP q )( H − z ) − M l . For the above formula, we have used that in L (cid:16) e − ǫ h x i L ( R ) , e ǫ h x i L ( R ) (cid:17) M l P q ( H − (Λ q + k )) − M l = M l P q (Λ q − (Λ q + k )) − M l = − k M l P q M l . Under obstacle perturbation, our main result shows that z = Λ q remains an essential singularity. Butit is not excluded that Λ q becomes also an isolated singularity coming from the perturbation of theholomorphic part Hol Λ q ( z ) .Our assumption (4.17) which is equivalent to (ii) does not allow this possibility. It is reasonable tothink that (4.17) is generic, for example in the sense that if z = Λ q becomes a isolated singularity of M l ( I + ikP q ) ˜ R l ( z ) M l then, under a small perturbation of the obstacle K , this singularity disappears.In particular, for l = γ , among the possible singularities of M l ( I + ikP q ) ˜ R l ( z ) M l there are theeigenvalues of the interior operator H − γK which has a discrete spectrum. Although it seems to be anopen question, we hope that if Λ q is an isolated eigenvalue of H − γK , then under a small perturbationof K , this eigenvalue moves to another value close to (but different from) Λ q .In order to simplify the statement of Theorem 2.1, let us introduced the following definition. Definition 4.7.
We will say that the obstacle K doesn’t produce an isolated resonance at Λ q if theproperty (ii) of Proposition 4.6 is satisfied. AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 15
Lemma 4.8.
Near k = 0 , Im( k ) > , Re( k ) > , there exists a holomorphic operator-valuedfunction K such that: I − ε ( l ) A lq ( ik ) ik ! = (cid:16) I − K ( k ) (cid:17) (cid:16) I − ε ( l ) Λ q (Λ q + k ) ik M l P q M l (cid:17) lim k → k ) >δ Im( k ) > ( I − K ( k )) = I − ε ( l )( A lq ) ′ (0)Π q , where δ > is fixed.Proof. The proof of this Lemma follows similarly to that of Proposition 3.6 (or of Lemma 4.1) of [4]where the same assumption (4.17) appears. We have A lq ( ik ) ik = A lq (0)(Λ q + k ) ik Λ q + A lq ( ik ) − Λ q + k Λ q A lq (0) ik = A lq (0)(Λ q + k ) ik Λ q + ( A lq ) ′ (0) + kR l ( ik ) , where R is a holomorphic operator valued function near k = 0 . Then, since A lq (0) is self-adjoint,for ik ∈ C \ R , we have:(4.21) I − ε ( l ) A lq ( ik ) ik ! = (cid:16) I − K ( k ) (cid:17) I − ε ( l ) A lq (0)(Λ q + k ) ik Λ q ! , with K ( k ) = ε ( l )( A lq ) ′ (0) I − ε ( l ) A lq (0)(Λ q + k ) ik Λ q ! − + ε ( l ) kR l ( ik ) I − ε ( l ) A lq (0)(Λ q + k ) ik Λ q ! − . For
Re( k ) > δ Im( k ) > , | k | sufficiently small and ν ( k ) > such that ν ( k ) = o (1) , | k | = o ( ν ( k )) as | k | tends to , we have:(4.22) k (cid:16) I − ε ( l ) A lq (0)(Λ q + k ) ik Λ q (cid:17) − k = sup λ j ∈ σ ( ε ( l ) A lq (0)) Λ q | k || ik Λ q − λ j (Λ q + k ) | ≤ | k || Re( k ) | ≤ r δ , (4.23) k [ ν ( k ) , + ∞ [ ( A lq (0)) (cid:16) I − ε ( l ) A lq (0)(Λ q + k ) ik Λ q (cid:17) − k ≤ | k | ν ( k ) − | Im k | ≤ | k | /ν ( k )1 − | k | /ν ( k ) , (4.24) s − lim | k |→ ]0 ,ν ( k )[ ( A lq (0)) = 0 . Then, combining the compactness of ( A lq ) ′ (0) with (4.24) and (4.22), we obtain: lim k → k ) >δ Im( k ) > ( A lq ) ′ (0) ]0 ,ν ( k )[ ( A lq (0)) (cid:16) I − ε ( l ) A lq (0)(Λ q + k ) ik Λ q (cid:17) − = 0 , and from (4.23) and (4.22), we deduce(4.25) lim k → k ) >δ Im( k ) > ( I − K ( k )) = I − ε ( l )( A lq ) ′ (0)Π q . This concludes the proof of Lemma 4.8 by using the relations (4.21), (4.25), (4.9) and (4.20). (cid:3)
Proof of Proposition 4.6
By definition of A lq , for z ∈ S δ = (cid:8) z ∈ C ; Im( z ) > δ | Re( z ) − Λ q | (cid:9) , δ > , and k = p z − Λ q , Im( k ) > , Re( k ) > , we have I − ε ( l ) A lq ( ik ) ik ! = (cid:18) I + ε ( l ) M l (cid:16) z − H − (cid:17) − M l (cid:19) . Then from (3.19) and Lemma 4.8, we deduce, for z = Λ q + k ∈ S δ :(4.26) (cid:16) I − K ( k ) (cid:17) (cid:16) I − ε ( l ) Λ q (Λ q + k ) ik M l P q M l (cid:17) (cid:16) I − ε ( l ) M l (cid:16) z − ˜ R l (0) (cid:17) − M l (cid:17) = I. By exploiting that M l M l = ε ( l ) V l = ε ( l )( H − − ˜ R l (0)) (cid:0) see (3.5), (3.2), (3.3), (3.17) (cid:1) , we have:(4.27) (cid:18) I − ε ( l ) Λ q (Λ q + k ) ik M l P q M l (cid:19) I − ε ( l ) M l (cid:18) z − ˜ R l (0) (cid:19) − M l ! = I − ε ( l ) M l (cid:18) z − ˜ R l (0) (cid:19) − M l + ε ( l ) Λ q (Λ q + k ) ik M l P q (cid:18) H − − z (cid:19) (cid:18) z − ˜ R l (0) (cid:19) − M l . Using that, in L (cid:16) e − ǫ h x i L ( R ) , e ǫ h x i L ( R ) (cid:17) , P q H − = (Λ q ) − P q , (4.27) equals(4.28) I − ε ( l ) M l ( I + ikP q ) (cid:16) z − ˜ R l (0) (cid:17) − M l = I − ε ( l ) M l ( I + ikP q ) (cid:16) zI + z ˜ R l ( z ) (cid:17) M l whose limit as S δ ∋ z −→ Λ q exists if and only if(4.29) lim S δ ∋ z −→ Λ q M l ( I + ikP q ) ˜ R l ( z ) M l exists.Then Proposition 4.6 follows from (4.25), (4.26), (4.27), (4.28) and (4.29). Remark 4.9.
In Lemma 4.8, the limit is for arg( k ) ∈ (0 , π − θ δ ) with θ δ = arctan δ , and then, inProposition 4.6, it is sufficient to take S δ = (cid:8) z ∈ C ; arg( z − Λ q ) ∈ (0 , π − θ δ ) (cid:9) .
5. R
EDUCTION TO T OEPLITZ OPERATORS WITH SYMBOL SUPPORTED NEAR THE OBSTACLE
In this section we will prove Proposition 4.5. To the operators V l , l = ∞ , γ defined on L ( R ) by (3.2), (3.3) we associate the operator(5.1) W l := 12 Z R x V l dx , defined on L ( R ) by (4.6)Our goal is to study the counting function of the compact non negative operators T lq := Λ q p q W l p q , l = ∞ , γ, where p q is the orthogonal projection onto ker ( H Landau − Λ q ) .First, we study properties of V l in L ( R ) . For q ∈ N , let us introduce a compact domain K ⊂ R which contains K and(5.2) E q ( K ) = (cid:8) f ∈ L ( R ) ∩ C ∞ ( R ); ( H − Λ q ) f = 0 on K (cid:9) . It is an infinite dimensional subspace of L ( R ) which contains all functions ( P q f ⊥ ⊗ χ ) when χ ∈ L ( R x ) ∩ C ∞ ( R x ) satisfies D x χ = 0 on I K , defined as for K by (3.10). AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 17
Proposition 5.1.
Fix K , K two compact domains of R , K ⊂ K ⊂ K with ∂K i ∩ ∂K = ∅ , i = 0 , . For l = ∞ , γ , there exists L q a finite codimension subspaces of E q ( K ) and C > such thatfor any f ∈ L q , (5.3) C h f, K f i L ( R ) ≤ h H f, ε ( l ) V l H f i L ( R ) ≤ C h f, K f i L ( R ) , l = ∞ , γ. Proof.
The proof of the lower bound in the Dirichlet case is inspired by the analog result in the 2Dcase (see Proposition 3.1 of [17]). By introducing the operator H + K , we have V ∞ := H − − ( H ∞ Ω ) − ⊕ V ∞ + V ∞ with V ∞ = H − − ( H + K ) − , V ∞ = ( H + K ) − − ( H ∞ Ω ) − ⊕ . Since the quadratic form associated to ( H + K ) coincide with Q ∞ Ω (cid:0) see (1.7) (cid:1) on C ∞ (Ω) (cid:0) identifiedwith (cid:8) u ∈ C ∞ ( R ) supported in Ω (cid:9)(cid:1) , then V ∞ is a non negative operator on L ( R ) .Moreover, for V ∞ , we have: V ∞ = H − K (cid:16) I − K ( H + K ) − K (cid:17) K H − . Then exploiting that K ( H + K ) − K is a compact operator in L ( L ( R )) , we deduce that, ona finite codimension subspace of L ( R ) , we have: H V ∞ H ≥ K . This implies the lower bound of (5.3) in the Dirichlet case. The other estimates (lower bound for l = γ and upper bounds) are consequences of Lemma 5.2 and Lemma 5.3 below. (cid:3) Lemma 5.2.
Fix q ∈ N . For l = ∞ , γ and K ⊂ K there exists T l Σ an elliptic pseudo differentialoperator, on L (Σ) , of order and L q a finite codimension subspaces of E q ( K ) such that for any f ∈ L q , h H f, V ∞ H f i L ( R ) = Λ q h f | K , f | K i L ( K ) − h f | Σ , T ∞ Σ f | Σ i L (Σ) . h H f, V γ H f i L ( R ) = −h f | Σ , T γ Σ f | Σ i L (Σ) . The above lemma is comparable to Lemma 4.2 of [10]. The proof, which is closely related to the2D case (see Subsection 6.3 below), exploits the expressions of V l in terms of Dirichlet-Neumann and
Robin-Dirichlet operators (cid:0) see (3.15) and (3.16) (cid:1) and their elliptic properties as pseudo differentialoperators on Σ (see Proposition 6.4). Moreover, for f satisfying (( ∇ A ) +Λ q ) f = 0 in K , there existsan elliptic pseudo differential operators R γq of order on L (Σ) such that ∂ A,γ Σ f = R γq ( f | Σ ) + F γq ( f ) with F γq a finite rank operator (see Lemma 6.5). In particular, for γ = 0 , ∂ AN f = R q ( f | Σ ) + F q ( f ) (cid:0) for more details, we also refer to Remark 3.12 of [10] (cid:1) .As in the proof of Lemma 3.14 of [10] which doesn’t depends on the even dimension of the space(see the end of Section 4 of [10]), we have Lemma 5.3.
Fix q ∈ N and K i , i = 0 , two compact domains of R , K ⊂ K ⊂ K , ∂K i ∩ ∂K = ∅ .Let T Σ be a non negative elliptic pseudo differential operator, on L (Σ) , of order . Then there exists M q a finite codimension subspaces of E q ( K ) and C > such that for any f ∈ M q , (5.4) C h f, K f i L ( R ) ≤ h f | Σ , T Σ f | Σ i L (Σ) ≤ C h f, K f i L ( R ) . Proof. of Proposition 4.5.
By definition of W l (cid:0) see (5.1) (cid:1) and exploiting the proof of Lemma 3.1, for f ⊥ ∈ ker ( H Landau − Λ q ) and any χ ∈ C ∞ c ( R x ) equal to on I K (cid:0) defined by (3.10) (cid:1) , we have h f ⊥ , T lq f ⊥ i L ( R ) = ε ( l ) Λ q Z R f ⊥ ( x , x )( H − χ H V l H χ H − )( f ⊥ ⊗ R )( x , x , x ) dx. For f ⊥ ∈ ker ( H Landau − Λ q ) , in L (cid:16) e ǫ
6. B
OUNDARY OPERATORS
In this section we recall how the method of layer potential allows to prove that the
Dirichlet-Neumann et Neumann-Dirichlet operators are pseudo differential operators on a surface and howLemma 5.2 and Lemma 5.3 follow. In presence of a constant magnetic field, these technics wasalready used in [16], [10] for even-dimensional cases.6.1.
Green kernel for ( ∇ A ) near the diagonal. Let G ( x, y ) , x, y ∈ R be the integral kernelof H − . It is related to H ( t, x, y ) , the heat kernel, by the formula: G ( x, y ) = Z + ∞ H ( t, x, y ) dt, where (cid:0) see e.g. [2] (cid:1) , for x = ( x , x , x ) = ( x ⊥ , x ) ∈ R × R , H ( t, x, y ) = 1 √ πt b π sinh( bt ) exp (cid:26) − ( x − y ) t − b bt ) | x ⊥ − y ⊥ | − i b x ⊥ ∧ y ⊥ (cid:27) , with | x ⊥ | = x + x , x ⊥ ∧ y ⊥ = x y − x y . Then we obtain: AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 19
Lemma 6.1.
The integral kernel of H − is smooth outside the diagonal and we have (6.1) G ( x, y ) = K ( x, y ) + O (1) as | x − y | → , with (6.2) K ( x, y ) ∼ e − ib x ⊥ ∧ y ⊥ π | x − y | ∼ π | x − y | as | x − y | → . Moreover, for x, y ∈ Σ , ( ∂ AN ) y G ( x, y ) satisfies the corresponding behavior as | x − y | tends to ,where ( ∂ AN ) y means that the differentiation is with respect to the variable y . More precisely, we have (6.3) ( ∂ AN ) y G ( x, y ) = ( ∂ AN ) y e − ib x ⊥ ∧ y ⊥ π | x − y | ! + O (1) as | x − y | → . Proof.
By the change of variables u = bt , we can rewrite G ( x, y ) as (6.4) G ( x, y ) = e − ib x ⊥ ∧ y ⊥ I ( x, y ) , I ( x, y ) := b (4 π ) Z + ∞ e − b (cid:20) ( x − y u +coth( u ) | x ⊥ − y ⊥ | (cid:21) u sinh( u ) du. Then Lemma 6.1 is a direct consequence of the following lemma.
Lemma 6.2. (i)
The function I ( x, y ) defined by (6.4) can be rewritten as I ( x, y ) = I ( x, y ) + I ∞ ( x, y ) , where a) I ∞ ( x, y ) = O (1) uniformly with respect to the variables x, y . b) The function I ( x, y ) satisfies for | x − y | ≪ (6.5) I ( x, y ) = 1(4 π ) | x − y | Z + ∞ b | x − y | e − u u du + O (1) . (ii) The function ( ∂ AN ) y G ( x, y ) satisfies for | x − y | ≪ (6.6) ( ∂ AN ) y G ( x, y ) = − i X j =1 ν j A j G ( x, y ) + ν · ( x − y ) e − ib x ⊥ ∧ y ⊥ π ) | x − y | Z + ∞ b | x − y | u e − u du + O (1) . The proof of Lemma 6.2 is of computational nature. Hence, for more transparency in the presenta-tion, it is differed in the Appendix. Now let us back to the proof of Lemma 6.1.Identities (6.1) and (6.2) follows immediately from (i) of Lemma 6.2 together with (6.4) and re-marking that R + ∞ u − e − u du = 4 R + ∞ e − v du = (4 π ) .Identity (6.3) follows from (ii) of Lemma 6.2 remarking firstly that R + ∞ u e − u du = (4 π ) ,and secondly that ( ∂ AN ) y e − ib x ⊥ ∧ y ⊥ π | x − y | ! = e − ib x ⊥ ∧ y ⊥ ν · ( x − y )4 π | x − y | − i X j =1 ν j A j G ( x, y ) . This concludes the proof of Lemma 6.1. (cid:3)
Boundary operators associated to ( ∇ A ) . According to the properties of G near the diag-onal, the following single-layer and double-layer potentials of a function f on Σ are well defined:(6.7) S f ( x ) := Z Σ f ( y ) G ( x, y ) dσ ( y ) , x ∈ R \ Σ , (6.8) D f ( x ) := Z Σ f ( y )( ∂ AN ) y G ( x, y ) dσ ( y ) , x ∈ R \ Σ (cid:0) see for instance [21] (cid:1) . Moreover, for x ∈ Σ we have the following limit relations:(6.9) lim z → x S f Ω ( z ) = S f Σ ( x ) = lim z → x S f K ( z ) , (6.10) lim z → x D f Ω ( z ) = − f Σ ( x ) + D f Σ , (6.11) lim z → x D f K ( z ) = 12 f Σ ( x ) + D f Σ , where(6.12) S f Σ ( x ) := Z Σ f ( y ) G ( x, y ) dσ ( y ) , x ∈ Σ , (6.13) D f Σ ( x ) := Z Σ f ( y )( ∂ AN ) y G ( x, y ) dσ ( y ) , x ∈ Σ , define compact operators on L (Σ) . More precisely, following the arguments of Section 7.11 of [21] (cid:0) see also Lemmas 3.2, 3.3 and 3.6 of [10] (cid:1) , S and D are pseudo differential operators, on Σ , of order ( − , and S is an elliptic self-adjoint operator on L (Σ) which is an isomorphism from L (Σ) onto H (Σ) . Moreover for ϕ ∈ C ∞ (Σ) and • = K, Ω , f • := S ( S − ϕ ) |• is the unique solution of(6.14) (cid:26) ( ∇ A ) f • = 0 in • f •| Σ = ϕ, and we have:(6.15) S ( ∂ AN f K ) = ( D −
12 ) ϕ, S ( ∂ AN f Ω ) = ( D + 12 ) ϕ. Inserting ∂ Σ = ∂ AN + γ above, we obtain:(6.16) S ( ∂ Σ f K ) = ( D + S γ −
12 ) ϕ, S ( ∂ Σ f Ω ) = ( D + S γ + 12 ) ϕ. Remark 6.3.
Due to the ellipticity of S and the compactness of D and S , the operators ( D + S γ ± ) are Fredholm operators and consequently there inverse exist on finite codimension spaces. In otherwords, there exists elliptic pseudo differential operators R ± of order , such that R ± ( D + S γ ± ) − I L (Σ) and ( D + S γ ± ) R ± − I L (Σ) are finite rank operators.Moreover, as in [10] (see Lemma 3.7 and Corollary 3.10), for all ε ∈ [ − , outside a finite subset (cid:0) − ε / ∈ σ ( S − ( D ± ) + γ ) (cid:1) , the operators ( D + S ( γ + ε ) ± ) are invertible on L (Σ) . On this way, as in the 2D case (cid:0) see Proposition 3.8 of [10] (cid:1) we can give the definition of the
Dirichlet-Robin and
Robin-Dirichlet operators introduced in (3.15) and (3.16) :
AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 21
Proposition 6.4. (i)
The interior (resp. exterior) Dirichlet-Robin operator DR K (resp. DR Ω ) de-fined by (6.17) DR K = S − (cid:18) D + S γ − (cid:19) , DR Ω = S − (cid:18) D + S γ + 12 (cid:19) , are first order elliptic pseudo differential operators on Σ . The interior (resp. exterior) Dirichlet-Neumann operator DN K (resp. DN Ω ) corresponds to γ = 0 . (ii) The interior (resp. exterior) Robin-Dirichlet operator RD K (resp. RD Ω ) defined on a finitecodimensional subspace of L (Σ) by (6.18) RD K = (cid:18) D + S γ − (cid:19) − S , RD Ω = (cid:18) D + S γ + 12 (cid:19) − S , are elliptic pseudo differential operators on Σ of order ( − . The interior (resp. exterior) Neumann-Dirichlet operator ND K (resp. ND Ω ) corresponds to γ = 0 . Proof of Lemma 5.2.
From (3.15) and (3.16), for f ∈ E q ( K ) (defined by (5.2)), we have: h H f, V ∞ H f i L ( R ) = h f | K , Λ q f | K i L ( K ) + h f | Σ , ∂ AN f i L (Σ) − h f | Σ , DN Ω ( f | Σ ) i L (Σ) , and h H f, V γ H f i L ( R ) = −h ∂ A,γ Σ f , ( RD K − RD Ω ) ∂ A,γ Σ f i L (Σ) . Moreover, based on the methods of Section 6.2, by considering the Green function associated tothe operator ( ∇ A ) + Λ q on K , we construct S q and D q two pseudo differential operators, on Σ , oforder ( − , with S q elliptic on L (Σ) but not necessarly invertible. These operators satisfy relationslike (6.15), (6.16). Then following the proofs of Lemma 3.11 and Remark 3.12 of [10], we obtain: Lemma 6.5.
There exists an elliptic pseudo differential operators R γq of order on L (Σ) and a finiterank operator F γq such that for f satisfying (( ∇ A ) + Λ q ) f = 0 in K , (6.19) ∂ A,γ Σ f := ∂ AN f + γf = R γq ( f | Σ ) + F γq ( f ) . Then we deduce Lemma 5.2 from Proposition 6.4 and Lemma 6.5 with T ∞ Σ = DN Ω − R q , T γ Σ = ( R γq ) ∗ ( RD K − RD Ω ) R γq .
7. A
PPENDIX
This appendix is devoted to the proof of Lemma 6.2. Constants in the O ( · ) are generic, namelychanging from a relation to another. (i) Let I ( x, y ) be the function defined by (6.4). Define I ( x, y ) and I ∞ ( x, y ) by (7.1) I ( x, y ) := b (4 π ) Z e − b (cid:20) ( x − y u +coth( u ) | x ⊥ − y ⊥ | (cid:21) u sinh( u ) du and (7.2) I ∞ ( x, y ) := b (4 π ) Z + ∞ e − b (cid:20) ( x − y u +coth( u ) | x ⊥ − y ⊥ | (cid:21) u sinh( u ) du. Since coth( u ) ≥ for u ≥ , then clearly I ∞ ( x, y ) = O (1) uniformly with respect to x , y . Thisgives (i) a) of Lemma 6.2.Now let us prove (i) b) . By using the change of variables u = 1 /v , the integral I ( x, y ) given by(7.1) verifies (7.3) (4 π ) b I ( x, y ) = Z + ∞ e − b | x − y | v v e − b | x ⊥− y ⊥| [ coth( v ) − v ] v sinh( v ) dv. It can be easily checked that for | x − y | ≪ , and uniformly with respect to v ∈ [1 , + ∞ [ , we have (7.4) e − b | x ⊥− y ⊥| [ coth( v ) − v ] v sinh( v ) = 1 + O (cid:18) v (cid:19) , v ≥ . By combining (7.3) and (7.4), we get for | x − y | ≪ (7.5) (4 π ) b I ( x, y ) = Z + ∞ e − b | x − y | v v dv + O (1) . Now (i) b) of Lemma 6.2 is a direct consequence of (7.5) using the change of variables u = b | x − y | v . (ii) The proof of this point is quite similar to that of the previous. Let G ( x, y ) and I ( x, y ) be thefunctions defined by (6.4). By a direct computation, it can be checked that (7.6) ( ∂ AN ) y G ( x, y ) = e − ib x ⊥ ∧ y ⊥ X j =1 ν j ∂ j I ( x, y ) − i X j =1 ν j A j G ( x, y ) , where the differentiation is with respect to the variable y . So to conclude, it suffices to investigate theintegral functions ∂ j I ( x, y ) = ∂ j (cid:0) I ( x, y ) + I ∞ ( x, y ) (cid:1) , j = 1 , , , where I ( x, y ) and I ∞ ( x, y ) arethe functions defined respectively by (7.1) and (7.2). Firstly, an easy computation show that we have (7.7) ∂ j I ∞ ( x, y ) = ( x j − y j ) O (1) , j = 1 , , . Secondly, by using for example the expression (7.3) of I ( x, y ) , it can be checked that (7.8) (4 π ) b ∂ j I ( x, y ) = b ( x j − y j )2 Z + ∞ v e − b | x − y | v coth( v ) e − b | x ⊥− y ⊥| [ coth( v ) − v ] v sinh( v ) dv, j = 1 , , and (7.9) (4 π ) b ∂ I ( x, y ) = b ( x − y )2 Z + ∞ v e − b | x − y | v e − b | x ⊥− y ⊥| [ coth( v ) − v ] v sinh( v ) dv. Similarly to the expansion (7.4), it can be proved that the functions h ( v ) := coth( v ) e − b | x ⊥− y ⊥| [ coth( 1 v ) − v ] v sinh( v ) and k ( v ) := e − b | x ⊥− y ⊥| [ coth( 1 v ) − v ] v sinh( v ) appearing respectively in the integrals (7.8) and (7.9) satisfy for | x − y | ≪ (7.10) h ( v ) = 1 + O (cid:18) v (cid:19) , k ( v ) = 1 + O (cid:18) v (cid:19) , v ≥ . This together with (7.8) and (7.9) give for j = 1 , , and | x − y | ≪ (7.11) (4 π ) b ∂ j I ( x, y ) = b ( x j − y j )2 "Z + ∞ v e − b | x − y | v dv + Z + ∞ e − b | x − y | v v dv O (1) . AGNETIC RESONANCES FOR EXTERIOR PROBLEMS 23
After the change of variables u = b | x − y | v , finally we get (7.12) (4 π ) b ∂ j I ( x, y ) = b ( x j − y j )2 " b | x − y | Z + ∞ b | x − y | u e − u du + 1 b | x − y | Z + ∞ b | x − y | e − u u du O (1) . Consequently, (6.6) of point (ii) follows from (7.6), (7.7) and (7.12). This concludes the proof of theLemma 6.2.Acknowledgments.The authors are grateful to G. Raikov for his continued support and helpful exhange of views. Thefirst author thanks the Mittag-Leffler Institut where this work was initiate with useful discussions withM. Persson and with G. Rozenblum.V. Bruneau was partially supported by ANR-08-BLAN-0228. D. Sambou is partially supported bythe Chilean Program
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