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RESONANCES AND SPECTRAL SHIFT FUNCTION FOR A MAGNETIC SCHRÖDINGER OPERATOR 9 ABDALLAHKHOCHMAN 0 0 2 n H 0 a Abstra t. We onsider the3vD S hrödinger operator with onstantmagneti (cid:28)eldaxnd 0 3 J subje ttoanele tri potential dependingonlyonthevariablealongthemagneti (cid:28)eld . H 0 4 Theoperator hasin(cid:28)nitelyHm0anyeigenvaluesofin(cid:28)nitemuVltip=li Oit(yh(exm1,bxe2d)di−edδ⊥inhxi3tis− δokn)- 1 tinuousspe trum. Weperturb bysmooths alarpotentials , δ⊥ >2, δk >1. We assume also that V and v0 have an analyti ontinuation, in the mag- ] neti (cid:28)elddire tion,in a omplexse toroutside a ompa tset. We de(cid:28)netheresonan es of P H =H0+V H .S distqor∈tioNns ofaRsxt3h.eWeiegesntuvadlyuetsh2eobifrqt+dhiesλtnriobnu-tsieolnfadnjeoairnta(rno,yp2er(cid:28)r)xaetodrroeabltaeiingeednvfarloumeofHby0,a2nbaqly+tiλ h for . Inaring enteredat withradiuses ,weestablishanupperbound,as r 0 V t tendsto , of the numberof resonan es. This upper bounddepends on thede ay of at a (x1,x2) m in(cid:28)nityonly in the dire tions . Finally, we dedu Hea,Hrepresentation of the derivative 0 ofthespe tralshiftfun tion(SSF)fortheoperatorpair( )intermsofresonan es. This [ representation justi(cid:28)es theBreit-Wigner approximationand implies a lo al tra e formula. 1 Mathemati s lassi(cid:28) ation: 35P25, 35J10, 47F05, 81Q10. v 0 Keywords: Ele tromagneti S hrödinger operator, Resonan es, Embedded eigenvalue, 8 Spe tral shift fun tion, Breit-Wigner approximation, Tra e formula. 9 1 . 1. Introdu tion 1 0 The resonan e theory for non-relativisti parti les satisfying the S hrödinger equation has 9 been developed following several approa hes. Among them we an mention the analyti dila- 0 : tion (see Aguilar-Combes [1℄) or the analyti distortion (see Hunziker [10℄) and meromorphi v ontinuation of the resolvent or of the s attering matrix (see Lax-Philips [14℄ and Vainberg i X [20℄). For S hrödinger operators with onstant magneti (cid:28)eld, the resonan es an be de- r (cid:28)ned by analyti dilation (only) with respe t to the variable along the magneti (cid:28)eld (see a Avron-Herbst-Simon [3℄, Wang [21℄, Astaburuaga-Briet-Bruneau-Fernández-Raikov [2℄) and by meromorphi ontinuation of the resolvent (see J.F.Bony-Bruneau-Raikov [4℄). The link between the resonan es and the spe tral shift fun tion (SSF) by the so- alled Breit-Wigner approximation has been developed in di(cid:27)erent situations. Su h a representation of the derivative of the spe tral shift fun tion related to the resonan es, implies tra e formulas. In the semi- lassi al regime we an mention Sjöstrand [18℄, [19℄, Petkov-Zworski [15℄, J.F.Bony-Sjöstrand [5℄, Bruneau-Petkov [6℄ and Dimassi-Zerzeri [8℄ for the S hrödinger operator and [12℄ for the Dira operator. In [4℄, J.-F.Bony, Bruneau and Raikov obtain a 3 Breit-Wigner approximation of the spe tral shift fun tion near a Landau level for the D S hrödinger operator with onstant magneti (cid:28)eld. For the last operator, under more general assumptions, Fernández-Raikov [9℄ studied the singularities of the spe tral shift fun tion at a Date: January 14, 2009. 1 2 ABDALLAHKHOCHMAN Landau level. These singularities has been analysed at eigenvalues of in(cid:28)nite multipli ity by Astaburuaga, Briet, Bruneau, Fernández and Raikov [2℄ for a magneti S hrödinger operator having ele tri potential depending (only) on the variable along the magneti (cid:28)eld. H 0 In this paper we onsider the magneti S hrödinger operator with an ele tromagneti (cid:28)eld introdu ed in [2℄. We suppose that the magneti (cid:28)eld is onstant and that the ele tri v x 0 3 potential depends only on the variable and is analyti outside a ompa t set. This operator isremarkablebe auseofthegeneri presen eofin(cid:28)nitely many eigenvaluesofin(cid:28)nite H H 0 0 multipli ity, embedded in the ontinuous spe trum of . We perturb the operator by a V x 3 smooth s alar potential analyti outside a ompa t set with respe t to the variable . The purpose of this work is to de(cid:28)ne the resonan es of the ele tromagneti S hrödinger H = H +V 0 operator for analyti perturbation outside a ompa t set in the third dire tion x H 3 . We de(cid:28)ne the resonan es for as the dis rete eigenvalues of the non-selfadjoint operator H θ Robtained from the magneti S hrödinger operator by agenHeral lass of omplex distortions of x3. In Se tion 3, we prove that the dis rete eigenvalues of θ are thezeros of aregularized det (·) 2 determinant whi h is independent of the distortion. This justi(cid:28)es the de(cid:28)nition of the resonan es. We al ulate the essential spe trum of the distorted operator to determine the se tor where we an de(cid:28)ne the resonan es. In Se tion 4, we establish an upper bound for the H r → 0 H 0 number of resonan es of in a domain of size near an embedded eigenvalue of . The se ond goal of this work is to obtain a Breit-Wigner approximation for the derivative of ξ(λ) H the spe tral shift fun tion related to the resonan es of the operator , as well as a lo al tra e formula (see Se tion 5). 2. Assumptions and results H 0 In this se tion, we summariBze=so(m0,e0s,pb)e tbra>l p0roperties of the 3D S hrödinger operator with onstant magneti (cid:28)eld , and subje t to a non- onstant ele tri (cid:28)eld E = −(0,0,v (x )) x 0′ 3 depending onlyonthevariable 3 (see[2℄). Wealsostatethemainresults. Let H = H ⊗I +I ⊗H , 0 0, 0, (2.1) ⊥ k ⊥ k I I L2(R ) L2(R2 ) where k and ⊥ are the identity operators in x3 and x1,x2 respe tively, 2 2 ∂ bx ∂ bx H := i − 2 + i + 1 −b, (x ,x ) ∈R2, 0, 1 2 (2.2) ⊥ (cid:18) ∂x1 2 (cid:19) (cid:18) ∂x2 2 (cid:19) b L2(R2) is the Landau Hamiltonian shifted by the onstant , self-adjoint in , and d2 H := − +v , x ∈ R. (2.3) 0,k dx23 0 3 v v (x ) 0 0 3 The operator isvth∈eLmu(ltRip)li ation operator by an one dimensional s alar potential . 0 ∞ We suppose that and satis(cid:28)es |v | = O(hx i δ0), 0 3 − (2.4) hxi = (1+|x|2)21 δ0 > 1 with and . Then using Weyl theorem, we have d2 σ (H )= σ (− ) =[0,+∞[. (2.5) ess 0,k ess dx2 3 MAGNETIC SCHRÖDINGER RESONANCES AND SSF 3 H 2bq q ∈ 0, INt:i=s w{e0l,l1k,n2o.w..n}that the spe trum of the operator ⊥ 2oqnbsists of the Landau levels , , aHndthemultipli ity2obfqe+a hλeigenvaqlu∈e N isinλ(cid:28)nite(see[3℄). Consequently, 0 the eigenvalues of have the forHm = − d2w+hvere(x ) and is an eigenvalue of the one dimensional S hrödinger operator 0,k dx23 0 3 . For simpli ity, throughout thearti le we suppose also that σ(H ) > −2b. 0, (2.6) inf k v > −2b H 2bq+λ, q ∈ N 0 0 ∗ Note that, (2.6) holds true if . The eigenvalues of , , are embedded [0,+∞[= ∪ [2bq,∞[ ∞q=0 in its ontinuous spe trum and are of in(cid:28)nite multipli ity. H = H + V V 0 Now, we introdu e theVp(exr)turbed operatorV ∈L (R3) where is the multipli ation ∞ operator by the potential . Assume that and satis(cid:28)es |V(x)| = O(hX i−δ⊥hx3i−δk), X = (x1,x2), (2.7) ⊥ ⊥ δ > 2 δ > 1 V v 0 with ⊥ and k .xWe suppose also that and have holomorphi extensions in the 3 magneti (cid:28)eld dire tion in the se tor C := {z ∈ C;| (z)| ≤ ǫ| (z)|, | (z)| ≥ R > 0}, 0 < ǫ < 1, ǫ,0 0 (2.8) Im Re Re for x ∈ C 3 ǫ,0 and satisfy respe tively (2.7) and (2.4) for . θ ∈ D ∩R D := {θ ∈ C; |θ|≤ r := ǫ } For ǫ with ǫ ǫ √1+ǫ2 , we denote H := (I ⊗U )H(I ⊗U 1)= H +V , θ θ θ− 0,θ θ ⊥ ⊥ where H := (I ⊗U )H (I ⊗U 1) = H ⊗I +I ⊗H (θ) (2.9) 0,θ ⊥ θ 0 ⊥ θ− 0,⊥ k ⊥ 0,k H (θ) = U H U 1 U and 0,k θ 0,k θ−H(see (3.2) for the de(cid:28)nition ofθ ∈θ).DWe will prouve in the next θ ǫ se tion that the operator has an analyti extension for . θ D+ := D ∩{θ ∈ C; (θ)≥ 0} q ∈ N r ∈ R 0 ǫ ǫ For (cid:28)xed in Im , and , we de(cid:28)ne Γ := 2br+(1+θ ) 2[0,+∞[ (2.10) r,θ0 0 − and S := Γ . (2.11) q,θ0 r,θ0 q<r[<q+1 H The spe trum of 0,θ0 is purely essential and we have σ(H ) = σ (H ) = 2bq+σ(H (θ )) (2.12) 0,θ0 ess 0,θ0 q[N(cid:0) 0,k 0 (cid:1) ∈ = Γ ∪ 2bq+σ (H (θ )) , q,θ0 disc 0, 0 q[N(cid:0) (cid:0) k (cid:1)(cid:1) ∈ σ (H (θ )) = σ (H )∪{z ,z ,...} σ (H ) disc 0, 0 disc 0, 1 2 disc 0, where k k , k denotes the dis rete spe trum of H z ,z ,... H (θ ) 0, 1 2 0, 0 k and are the omplex eigenvalues of k . In the following, we assume that σ (H )= {λ} λ disc 0, k . Note that is ne essarily simple. H H θ 0,θ The essential spe trum of oin ides with that of . We prove also that the dis rete H S = S θ D+ θ , θ spe trum of θ in θH Sq∈NHq,θ is independSent∩ofS in ǫ (i.e. for two values 1 2, the dis rete spe trum of θ1 and θ2 oin ide on θ1 θ2), (see Proposition 3.2). This justi(cid:28)es the following de(cid:28)nition. 4 ABDALLAHKHOCHMAN H S H De(cid:28)nition 2.1. The resonan es of in θ0 are the dis rete eigenvalues of θ0. The multi- z 0 pli ity of a resonan e is de(cid:28)ned by 1 (z ) := (z−H ) 1dz, (2.13) mult 0 rank2iπ Z θ0 − Γ0 Γ z (H) 0 0 where is a small positively oriented ir le entered at . We will denote Res the set of resonan es. H {z ∈ C; (z) < 0} Remark 2.1. The resonan es of in Re are the real dis rete eigenvalues H of . PSfrag repla ements (z) Im Ω q H Resonan es of H e.v. of θ0 λ 0 2b+λ 2b 2bq+λ 2bq (z) ◦ ◦ ◦ ◦ Re ⋆ ⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆ ⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆⋆⋆ ⋆⋆Γ⋆⋆⋆⋆◦⋆⋆⋆ ⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆◦⋆⋆⋆⋆⋆⋆⋆ ⋆ Γ q,θ0 1,θ0 Γ 0,θ0 S θ0 S Fig.1. The set θ0 r → 0 H Ω q Now, we state(ra,n2ru)pper bound as 2bq+oλn tqh∈e nNumber of resonan es of in a ring in ∗ with radiuses and entered at , (cid:28)xed. V v 0 Theorem 2.1. [Upper bound℄ Suppose that and satisfy the above hypotheses. Then r > 0 ν > 0 0< r < r 0 0 there exist and , su h that, for any , #{z ∈ (H)∩Ω ;r < |z−2bq−λ|< 2r} = O(n (r,νp Wp )|lnr|), q + q q (2.14) Res whereW = supx3∈Cǫ,0|hx3iδkV|, pq istheorthogonalproje tionontoHq := ker(H0,⊥−2bq)and n (r,p Wp ) r + q q is the ounting fun tion of the eigenvalues larger than of the Toeplitz operator pqWpq n+(r,pqWpq) = O(r−2/δ⊥) . In parti ular, under our assumption we have always . n (r,p Wp ):= 1 (p Wp ) + q q (r,+ ) q q The ounting fun tion W rank ∞ satis(cid:28)es asymptoti relations depending on the de ay of at in(cid:28)nity. The following three lemmas give an upper bound n (r,p Wp ) W + q q of in the ase power-like de ay, exponential de ay, or ompa t support of , respe tively. For more pre ise results on erning the asymptoti properties, we refer to the ited theorem. MAGNETIC SCHRÖDINGER RESONANCES AND SSF 5 U ∈ L (R2) ∞ Lemma 2.1. (Theorem 2.6 of [16℄) Let the fun tion satisfy the estimate U(X )≤ ChX i α, X ∈ R2, − ⊥ ⊥ ⊥ α> 0 q ∈ N for some . Then for ea h , we have n (r,p Up )= O(r 2/α). + q q − U ∈L (R2) ∞ Lemma 2.2. (Theorem 2.1 of [17℄) Let . Assume that lnU(X ) limsup ⊥ < 0, X ∈ R2, |X⊥|→∞ |X⊥|2β ⊥ β > 0 ln(u) = −∞ u≤ 0 q ∈N for some (with the onvention if ). Then for ea h , we have n (r,p Up ) = O(ϕ (r)) + q q β 0 < r < e 1 − where, for , 1 |lnr|β 0< β < 1, if ϕβ(r):=  |lnr| β = 1, (2.15)  (ln|lnr|) 1|lnr| if β > 1. − if  U ∈ L (R2) U ∞ Lemma 2.3. (Theoremq2∈.4Nof [17℄) Let . Assume that the support of is ompa t. Then for ea h , we have n (r,p Up )= O(ϕ (r)), + q q ∞ 0 < r < e 1 − where, for , ϕ (r):= (ln|lnr|) 1|lnr|. − ∞ (H,H ) ξ(λ) 0 Now, we study the spe tralHsh,iHft fun tion (SSF) for thDe p(aRi)r . The SSF for a 0 ′ pair of self-adjoint operators ( ) is a distribution in whose derivative is ξ :f ∈ C (R) 7−→ − (f(H)−f(H )). ′ 0∞ 0 (2.16) tr |V|21(H0+i)−1 Iξn(λo)ur ase, L1 (R)is in the Hilbert-S hmidt lass, (2.16) is well de(cid:28)ned and the SSF is a fun tion in loc . H S Wewillseefurtherthattheresonan esof in θ0 arethezerosoftheholomorphi extension of z ∈ {z ∈ C, z > 0} 7−→ D(z) = ((H −z)(H −z) 1) 2 0 − Im det S det into θ0 (see (3.1) for the de(cid:28)nition of 2). Thus in order to obtain a link between the SSF and the resonan es, it will be onvenient to introdu e the regularized spe tral shift fun tion 1 ξ (ν)= lim arg (H −ν −iε)(H −ν −iε) 1 , 2 2 0 − (2.17) π ε 0+ det → (cid:0) (cid:1) whose derivative is the following distribution (see [4℄) d ξ : f ∈ C (R)7−→ − f(H)−f(H )− f(H +εV)| . (2.18) 2′ 0∞ tr(cid:18) 0 dε 0 ε=0(cid:19) We will dedu e the properties of the SSF from those of the regularized SSF using the relation 1 ξ = ξ + V (H −z) 2 . (2.19) ′ 2′ πIm tr θ 0,θ − (cid:0) (cid:1) 2bq +λ We represent now, the derivative of the spe tral shift fun tion near as a sum of a harmoni measurerelatedtotheresonan esandtheimaginary partofaholomorphi fun tion. 6 ABDALLAHKHOCHMAN Ω ⊂⊂Ω C\{0} Let be open relatively ompa t subsets of . We assume that these sets are r Ω indepenedent of and that is simply onne ted. Also assume that the interse tions between Ω R I and is a non-empty inteerval . e V v 0 Theorem 2.2. [Breit-Wigner approximation℄ We suppose and satisfy the above Ω ⊂⊂Ω I g Ω hypothesis. For and as above, there exists a fun tion holomorphi in , su h that µ ∈ 2bq+λ+rI for e , we have 1 µ−2bq−λ − w ξ′(µ) = g′( ,r)− Im − δ(µ−w) πrIm r π|µ−w|2 X X w∈Res(H)∩2bq+λ+rΩ w∈Res(H)∩2bq+λ+rI w=0 Im 6 g(z,r) where satis(cid:28)es the estimate 2 g(z,r) = O(n+(r,νpqWpq)|lnr|+n1(r/ν)+n2(r/ν)) = O(|lnr|r−δ⊥), ν > 0, (2.20) 0< r < r z ∈eΩ ne, p = 1, 2, 0 p uniformly with respe t to and , with de(cid:28)ned by n (r) := pqWpq1 e(p Wpe) p, r > 0. (2.21) p (cid:13) r [0,r] q q (cid:13) (cid:13) (cid:13)p e (cid:13) (cid:13) (cid:13) (cid:13) k·k (p = 1) (p = 2) p Here, stands for the tra e- lass norms and Hilbert-S hmidt norms . W Using [4, Corollary1℄, for de(cid:28)ned above satisfying the assumption of Lemma 2.1 with α ≥ 2 , we have 2 np(r) = O(r−α), p = 1,2. (2.22) e W = U Finally, if the assumption of Lemma 2.2 or 2.3 hold for , we have n (r) = o(ϕ (r)) r ց 0, p β (2.23) ϕβ(r) e the fun tion being de(cid:28)ned in Lemma 2.2 or 2.3. As in [15℄, [6℄ or [4℄ and repeating the arguments used in the proof of [4, Corollary 3℄, we dedu e from Theorem 2.1-2.2 the following theorem Ω ⊂⊂ Ω f Theorem 2.3. [Tra e formula℄ Let be as in Theorem 2.2. Suppose that is Ω φ ∈ C (Ω ∩ R) φ = 1 Ω ∩ R holomorphi on a neighborhood of aned that 0∞ satis(cid:28)es near . Then, under the assumptions of Theorem 2.2, we have the following tra e formula e H −2bq−λ H −2bq−λ w−2bq−λ 0 (φf)( )−(φf)( ) = f( )+E (r) f,φ tr(cid:18) r r (cid:19) r X e w (H) 2bq+λ+rΩ ∈Res ∩ with |E (r)|≤ M sup{|f(z)| : z ∈ Ω\Ω, z ≤ 0}×N (r), f,φ φ q Im e 2 Nq(r) = n+(r,νpqWpq)|lnr|+ n1(r/ν)+ n2(r/ν) = O(|lnr|r−δ⊥), Mφ where and depends φ only on . e e MAGNETIC SCHRÖDINGER RESONANCES AND SSF 7 3. Definition of resonan es via distortion analyti ity In this se tion, we start with the de(cid:28)nition oRf the deformation for the ele tromagneti S hrödinger operator by analyti distortion on x3. We al ulate the essential spe trum H H θ θ of the distorted S hrödinger operator . We prove that the dis rete eigenvalues of are independentofthedistortion, thisjusti(cid:28)esthede(cid:28)nitionofaresonan easadis reteeigenvalue H H θ of the distorted operator . We will also prove that the resonan es of repeated with their det (I + A) 2 multipli ity oin ide with the zeros of a regularized determinant de(cid:28)ned for a A Hilbert-S hmidt operator by (I +A) := det((I +A)e A), 2 − (3.1) det (see Krein [13℄). Let us now introdu e the one-parameter family of unitary distortions in the x 3 magneti (cid:28)eld dire tion : 1 U f(x)= J2 f(φ (x)), θ ∈ R, f ∈ S(R), (3.2) θ φθ(x) θ φ (x) = x+θg(x) g : R7−→ R J = det(I+θg (x)) where θ , is a smooth fun tion and φθ(x) ′ is the φ (x) g θ Ja obian of . We suppose that satis(cid:28)es the assumption |g (x)| < 1, x R ′ (Ag) (i) sgu(px)∈= 0, [−R0,R0], ( ( )),  (ii) g(x) = x, in the ompa t set K(⊃ [−Rsee,R2.]8). 0 0 (iii) outside a ompa t set  We re all that H := (I ⊗U )H(I ⊗U 1) = H +V . θ θ θ− 0,θ θ ⊥ ⊥ From (2.9) and using Kato's theorem [11, Theorem 4.5.35℄ we have the following (see also Hunziker [10℄ for S hrödinger operator and [12, Se tion 3℄ for the Dira operator). V Proposition 3.1. We suppose that the potential satis(cid:28)es all the assumptions of Se tion 2. Then we have θ ∈ D 7−→ H = H +V ǫ θ 0,θ θ (i) σ (H ) =σ (H ) = 2bNis+aσn(Hanaly(θti) ).family of type A. ess θ ess 0,θ 0, (ii) k H (θ) 0, Lemma 3.1. The essential spe trum of k is µ σ (H (θ))= ∈ C; µ ∈ [0,+∞[ . (3.3) ess 0,k n(1+θ)2 o The rest of the spe trum is σ (H )∪{z ,z ,...}, disc 0, 1 2 k σ (H ) H z ,z ,... disc 0, 0, 1 2 where k denotes the dis rete spe trum of k and are the omplex eigen- H (θ) 0, values of k . σ (H ) ∪ {z ,z ,...} H (θ) disc 0, 1 2 0, Remark 3.1. The part k of the spe trum of k orrespond to I ⊗H (θ) 0, eigenvalues with in(cid:28)nite multipli ity of ⊥ k . q ∈ N Ω 2bq+λ q In the following we (cid:28)x and a ompa t set entered at su h that Ω ∩σ (H ) = {2bq+λ}. q ess θ (3.4) Repeatingargumentsintheproofof[4,Proposition1℄,[4,Lemma1℄andusingtheresolvent equation (H −z) 1 = (H −v −z) 1 I −v (H −z) 1 , 0 − 0 0 − 0 0 − (cid:0) (cid:1) we obtain the following lemma. 8 ABDALLAHKHOCHMAN V(H − z) 1 ∂ (V(H − z) 1) {z ∈ 0 − z 0 − LCe;mm(za)3>.20.}The operators and S are holomoSrphi on 2 1 Im with values intheHilbert-S hmidt lass andinthe tra e lass respe tively. V (H −z) 1 ∂ (V (H −z) 1) z ∈ θ 0,θ − z θ 0,θ − Lemma 3.3. The operators and are holomorphi for Ω \{2bq+λ} S S q 2 1 with values in the Hilbert-S hmidt lass and in the tra e lass respe tively. Ω z 7−→ q Proof. A ording to (i) of Proposition 3.1 and to the de(cid:28)nition of , the fun tion V (H −z) 1 z ∈ Ω \{2bq +λ} θ 0,θ − q is analyti for . Moreover, from the resolvent equation, for z ∈Ω \{2bq +λ} q , we have V (H −z) 1 = V (H −i) 1 1+(H −H +z−i)(H −z) 1 . θ 0,θ − θ 0 − 0 0,θ 0,θ − (3.5) (cid:0) (cid:1) p H := (H −2bq) q q 0, If we denote by the orthogonal proje tion onto ker ⊥ we have (H −z) 1 = p ⊗(H (θ)+2bq−z) 1, z ∈ Ω \{2bq +λ}. 0,θ − q 0, − q (3.6) Xq N k ∈ Sin e H −H = I ⊗(H −H (θ)) 0 0,θ 0, 0, ⊥ k k (H −H )(H −z) 1 0 0,θ 0,θ − the operator is bounded. Consequently, a ording to Lemma 3.2 an(cid:3)d equation (3.5), we obtain the lemma. Let us now introdu e the fun tion z ∈Ω \{2bq+λ} −→ d (z) = (I +T (z)), q θ 2 V,θ (3.7) det T (z) = V (H −z) 1 d (z) V,θ θ 0,θ − θ with . The determinant is well de(cid:28)ned a ording to Lemma 3.3 V v H Ω 0 q Proposition 3.2. Let and as in Se tion 2. The resonan es of in are the zeros of d (z) = (I +T (z)) Ω \{2bq +λ} θ 2 V,θ q the regularized determinant det in , and are independent θ ∈D Ω ∩σ (H ) = {2bq+λ} ǫ q ess θ on su h that . z f(z) z z 0 0 If is a resonan e, there exists a holomorphi fun tion , for lose to , su h that f(z ) 6= 0 0 and (I +T (z)) = (z−z )l(z0)f(z), 2 V,θ 0 det 0 < l(z )= (z ) (z ) 0 0 0 with mult where mult is the multipli ity of the resonan e de(cid:28)ned by (2.13). H Ω \{2bq +λ} 0,θ q Proof. Sin e the operator has no spe trum in , we have H −z = I +V (H −z) 1 (H −z). θ θ 0,θ − 0,θ (3.8) (cid:0) (cid:1) z ∈ Ω \{2bq +λ} H q Then, if is a resonan e of whi h is by de(cid:28)nition a dis rete eigenvalue of H d (z) = (I +V (H −z) 1) = (I +T (z)) θ θ 2 θ 0,θ − 2 V,θ , the determinant det det vanishes. A Let us re all that, if is a bounded operator and B is a tra e lass operator on some det(I +AB) = det(I +BA) A separable Hilbert spa e, we have . Moreover, for bounded and B Hilbert-S hmidt, we have (I +AB)= (I +BA). 2 2 (3.9) det det d (z) = det (I+T (z)) det (I+T (z)) = (I+V(H − θ 2 V,θ 2 V,0 2 0 zT)he1n), theθfu∈n Rti,on z > 0 oin idewith det θ ∈ D − ǫ for Im and by uniqueness of the extension, it is independent on . H d (z) θ ∈ D θ ǫ Sin e the resonan es of are the zero of the resonan es are independents on . MAGNETIC SCHRÖDINGER RESONANCES AND SSF 9 z d (z) l(z ) d (z) = (z − 0 θ 0 θ In a neighborhood of a zero of with multipli ity , we write z )l(z0)G(z), G(z) z G(z ) 6= 0 0 0 0 where is a holomorphi fun tion in a neighborhood of with . l (z) 0 Then, by the de(cid:28)nition of , 1 l (z) = ∂ ln (1+T (z))dz, 0 z 2 V,θ 2iπ Z det Γ Γ z 0 where is a small positively oriented ir le entered at . Further, we have ∂ lndet(1+T(z)) = (1+T(z)) 1∂ T(z) , z ∈ Ω z − z q tr (cid:0) (cid:1) T(z) S 1 for any operator-valued holomorphi fun tion in the tra e lass . Therefore, ∂ ln (1+T (z)) = (1+T (z)) 1∂ T (z) − (∂ T (z)). z 2 V,θ V,θ − z V,θ z V,θ det tr tr (cid:0) (cid:1) ∂ T (z) Γ z V,θ A ording to Lemma 3.3, is holomorphi in the tra e lass, then its integral on vanishes and (3.8) yields 1 l (z) = (H −z) 1V (H −z) 1 dz 0 θ − θ 0,θ − 2iπ Z tr Γ (cid:0) (cid:1) 1 = − (H −z) 1−(H −z) 1 dz θ − 0,θ − 2iπ Z tr Γ (cid:0) (cid:1) 1 = (z−H ) 1−(z−H ) 1 dz θ − 0,θ − rank2iπ Z Γ(cid:0) (cid:1) 1 = (z−H ) 1dz. θ − rank2iπ Z Γ In the two latter equ(Halitie−s,zw)e1haveΓused that the tra e of the proje tor oΩin ide with its ran(cid:3)k 0,θ − q and the integral of on vanishes sin e it is holomorphi in . 2bq+λ 4. Upper bound for the number of resonan es near Ω q In this se tion, we establish an upper bound on the number of resonan es in a ring of 2bq +λ z ∈ Ω z = 2bq +λ+η η q entered at (see (3.4)). For , we write where is a omplex number in a domain entered at 0 and 0 < r < |η|. Let W = supx3∈Cǫ,0|hx3iδkV|. There M(x) exists a bounded fun tion su h that V(x) = W(X )hx i 2δ3M(x), δ = δ /2. 3 − 3 ⊥ for k Ω q A ording to the previous se tion, the resonan es in an be identi(cid:28)ed with the points z ∈Ω d (z) = det (I +T (z)) q θ 2 V,θ where the determinant vanishes. Using (3.9), we have d (z) = (I +T (z)) θ 2 V,θ det with (4.1) TV,θ(z) = W21Mθhx3i−θδ3(H0,θ −z)−1W21hx3i−θδ3 where Mθ = M(X⊥,φθ(x3)) and hx3iθ := Uθhx3iUθ−1 = (1+(φθ(x3))2)12. z > 0 Using spe tral theorem, for Im , we an write (H −z) 1 = p ⊗(H (θ)−z+2bj) 1. 0,θ − j 0, − (4.2) Xj N k ∈ 10 ABDALLAHKHOCHMAN 2bq+λ T (z) V,θ In order to study the resonan es near , we split into two parts: T (z) = T +T+, V,θ J−,θ J,θ T (z) = T T+ = T J > q kT+k< w1herekTJ−,θ+k< 1Pj≤J j,θ and J,θ hPj>J j,θ for su(cid:30) iently large su h that J,θ 8 and J,0 8 (for that weusethe -pseudo-di(cid:27)erential al ulusand thespe traltheorem). Here, Tj,θ = MθBj ⊗hx3i−θδ3Rj,θhx3i−θδ3, Bj = W21pjW21 Rj,θ = (H0, (θ)−z+2bj)−1 hx3i−δ3Rj,0hx3i−δ3 with and k . The operator is of lass tra e (see [2℄). B j Further, let us de ompose the self-adjoint operator into a tra e- lass operator whose ε/2 ε > 0 r norm is bounded by for some and an operator of (cid:28)nite-rank independent on , namely B = B 1 (B )+B 1 (B ). j j [0,ε/2] j j ]ε/2,+ [ j (4.3) ∞ j 6= q Then, for , we have T = M B 1 (B )⊗hx i δ3R hx i δ3 +M B 1 (B )⊗hx i δ3R hx i δ3 j,θ θ j [0,ε/2] j 3 −θ j,θ 3 −θ θ j ]ε/2,+ [ j 3 −θ j,θ 3 −θ ∞ = T< +T>. j,θ j,θ T p (θ) H (θ)− q,θ 0, λ Letusnowpan(aθl)y·z=ethh·,eψteirψm . Dψen=oteUby1ψk thψespe tralproje tionontoKer( k ). We have k θ¯ θ with θ θ− and is an eigenfun tion satisfying H0, ψ =λψ, kψkL2(R) = 1, ψ = ψ¯ R. k on η = z−2bq−λ Then we have, for T =M B ⊗hx i δ3R p (θ)hx i δ3 +M B ⊗hx i δ3R (I −p (θ))hx i δ3 q,θ θ q 3 −θ q,θ 3 −θ θ q 3 −θ q,θ 3 −θ k k 1 =− τ +T˜ , q q,θ η with τq = MθBq ⊗hx3i−θδ3pk(θ)hx3i−θδ3. We also have T˜q,θ = MθBq1[0,ε/2](Bq)⊗hx3i−θδ3Rq,θ(I −p (θ))hx3i−θδ3 k + MθBq1]ε/2,+ [(Bq)⊗hx3i−θδ3Rq,θ(I −p (θ))hx3i−θδ3 ∞ k = T˜< +T˜>. q,θ q,θ A>(z) = T˜> + T>, A<(z) = T˜< + T< + T+. We denote by q,θ j6=q, j≤J j,θ and q,θ j6=q, j≤J j,θ J,θ P P Then, we have 1 T (z) = − τ +A>(z)+A<(z). V,θ q (4.4) η τ rν 1 q − We de ompose the operator into a tra e- lass operator whose norm is bounded by for ν > 0 and an operator of (cid:28)nite-rank: τq = MθBq1[0,rν−1](Bq)⊗hx3i−θδ3p (θ)hx3i−θδ3 +MθBq1]rν−1,+ [(Bq)⊗hx3i−θδ3p (θ)hx3i−θδ3 k ∞ k = τ +τ . q,1 q,2 ε ν > 0 If we take su(cid:30) iently small and su(cid:30) iently large, we have the two following lemmas