Maria Angélica Astaburuaga

Philippe Briet
email: briet@cpt.univ-mrs.fr

Vincent Bruneau
email: vbruneau@math.u-bordeaux1.fr

Claudio Fernandez
email: cfernand@mat.puc.cl

Georgi Raikov
email: graikov@mat.puc.cl

Í A Ang

Élica Astaburuaga

AND GEORGI RAIKOV Dedicated to Claudio Fern Ández

Vesselin Petkov

Dynamical resonances and SSF singularities for a magnetic Schroedinger operator

Keywords: 2000 AMS Mathematics Subject Classification: 35P25, 35J10, 47F05, 81Q10 magnetic Schrödinger operators, resonances, Mourre estimates, spectral shift function

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

In the present article we consider a magnetic Schrödinger operator H which, from a physics point of view, is the quantum Hamiltonian of a 3D non-relativistic spinless quantum particle subject to an electromagnetic field (E, B) with electric component E = -(0, 0, v ′ 0 ) where v 0 is a scalar potential depending only on the variable x 3 , and magnetic component B = (0, 0, b) where b is a positive constant. From a mathematical point of view this operator is remarkable because of the generic presence of infinitely many eigenvalues of infinite multiplicity, embedded in the continuous spectrum of H. These eigenvalues have the form 2bq + λ, q ∈ Z + := {0, 1, 2, . . .}, where 2bq, q ∈ Z + , 1 are the Landau levels, i.e. the infinite-multiplicity eigenvalues of the (shifted) Landau Hamiltonian, and λ is a simple eigenvalue of the 1D operator -d 2 dx 2 +v 0 (x). We introduce the perturbed operator H + κV where V is a H-compact multiplier by a real function, and κ ∈ R is a coupling constant, and study the transition of the eigenvalues 2bq + λ, q ∈ Z + , into a "cloud" of resonances which converge to 2bq + λ as κ → 0. In order to perform this analysis, we assume that V is axisymmetric so that the operator H + κV commutes with the x 3 -component of the angular-momentum operator L. In this case H + κV is unitarily equivalent to the orthogonal sum ⊕ m∈Z (H (m) + κV ) where H (m) is unitarily equivalent to the restriction of H onto Ker (Lm), m ∈ Z. This allows us to reduce the analysis to a perturbation of a simple eigenvalue 2bq + λ of the operator H (m) with fixed magnetic quantum number m. We apply two different approaches to the definition of resonances. First, we suppose that v 0 and V are analytic in x 3 , and following the classical approach of Aguilar and Combes [START_REF] Aguilar | A class of analytic perturbations for one-body Schrödinger Hamiltonians[END_REF], define the resonances as the eigenvalues of the dilated non-self-adjoint operator H(θ) + κV θ . We obtain an asymptotic expansion as κ → 0 of each of these resonances in the spirit of the Fermi Golden Rule (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF]Section XII.6]), and estimate the time decay of the resonance states. A similar relation between the smallcoupling-constant asymptotics of the resonance, and the exponential time decay of the resonance state has been established by Herbst [START_REF] Herbst | Exponential decay in the Stark effect[END_REF] in the case of the Stark Hamiltonian, and later by other authors in the case of various quantum Hamiltonians (see e.g. [START_REF] Skibsted | On the evolution of resonance states[END_REF], [START_REF] Ferrari | On the exponential decay of magnetic Stark resonances[END_REF], [START_REF] Asch | Sojourn time for rank one perturbations[END_REF]). Our other approach is close to the time dependent methods developed in [START_REF] Soffer | Time dependent resonance theory[END_REF] and [START_REF] Costin | Resonance theory for Schrödinger operators[END_REF], and, above all, to the recent article by Cattaneo, Graf and Hunziker [START_REF] Cattaneo | A general resonance theory based on Mourre's inequality[END_REF], where the dynamic estimates of the resonance states are based on appropriate Mourre estimates [START_REF] Mourre | Absence of singular continuous spectrum for certain self-adjoint operators[END_REF]. We prove Mourre type estimates for the operators H (m) , which might be of independent interest, and formulate a theorem on the dynamics of the resonance states which can be regarded as an application of the general abstract result of [START_REF] Cattaneo | A general resonance theory based on Mourre's inequality[END_REF]. Both our approaches are united by the requirement that the perturbation V satisfies the Fermi Golden Rule for all embedded eigenvalues for the operators H (m) , m ∈ Z. We establish several results which show that the set of such perturbations is dense in various topologies compatible with the assumptions of our theorems on the resonances of H + κV . Further, we cancel the restriction that V is axisymmetric but suppose that it decays fast enough at infinity, and has a definite sign, introduce the Krein spectral shift function (SSF) for the operator pair (H + V, H), and study its singularity at each energy 2bq + λ, q ∈ Z + , which, as before, is an eigenvalue of infinite multiplicity of the unperturbed operator H. We show that the leading term of this singularity can be expressed via the eigenvalue counting function for compact Berezin-Toeplitz operators. Using the wellknown results on the spectral asymptotics for such operators (see [START_REF] Raikov | Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips[END_REF], [START_REF] Raikov | Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials[END_REF]), we obtain explicitly the main asymptotic term of the SSF as the energy approaches the fixed point 2bq + λ for several classes of perturbations with prescribed decay rate with respect to the variables on the plane perpendicular to the magnetic field. It is natural to associate these singularities of the SSF to the accumulation of resonances to these points because it is conjectured that the resonances are the poles of the SSF. This conjecture is justified by the Breit-Wigner approximation which is mathematically proved in other cases (see for instance [START_REF] Petkov | Breit-Wigner approximation and the distribution of resonances[END_REF], [START_REF] Petkov | Semi-classical estimates on the scattering determinant[END_REF], [START_REF] Bruneau | Petkov Meromorphic continuation of the spectral shift function[END_REF], [START_REF] Bony | Resonances and spectral shift function near Landau levels[END_REF]). The article is organized as follows. In Section 2 we summarize some well-known spectral properties of the operators H and H (m) and their perturbations, which are systematically exploited in the sequel. Section 3 is devoted to our approach based on the dilation analyticity, while Section 4 contains our results obtained as corollaries of appropriate Mourre estimates. In Section 5 we discuss the density in suitable topologies of the sets of perturbations V for which the Fermi Golden Rule holds true for every embedded eigenvalue 2bq + λ of the operator H (m) , m ∈ Z. Finally, the asymptotic analysis of the SSF near the points 2bq + λ can be found in Section 6. We dedicate the article to Vesselin Petkov with genuine admiration for his most significant contributions to the spectral and scattering theory for partial differential operators. In particular, we would like to mention his keystone results on the distribution of resonances, and the Breit-Wigner approximation of the spectral shift function for various quantum Hamiltonians (see [START_REF] Petkov | Breit-Wigner approximation and the distribution of resonances[END_REF], [START_REF] Petkov | Semi-classical estimates on the scattering determinant[END_REF], [START_REF] Bruneau | Petkov Meromorphic continuation of the spectral shift function[END_REF]), and, especially, his recent works on magnetic Stark operators (see [START_REF] Dimassi | Spectral shift function and resonances for non-semi-bounded and Stark Hamiltonians[END_REF], [START_REF] Dimassi | Resonances for magnetic Stark Hamiltonians in two-dimensional case[END_REF]). These articles as well as many other works of Vesselin have strongly influenced and stimulated our own research.

Preliminaries

2.1.

In this subsection we summarize some well-known facts on the spectral properties of the 3D Schrödinger operator with constant magnetic field B = (0, 0, b), b = const. > 0. More details could be found, for example, in [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF] or [START_REF] Bruneau | Spectral shift function in strong magnetic fields[END_REF]Section 9].

Let H 0 := H 0,⊥ ⊗ I + I ⊥ ⊗ H 0,
where I and I ⊥ are the identity operators in L 2 (R x 3 ) and L 2 (R 2 x 1 ,x 2 ) respectively,

H 0,⊥ := i ∂ ∂x 1 - bx 2 2 2 + i ∂ ∂x 2 + bx 1 2 2 -b, (x 1 , x 2 ) ∈ R 2 ,
is the Landau Hamiltonian shifted by the constant b, and

H 0, := - d 2 dx 2 3 , x 3 ∈ R.
The operator H 0,⊥ is self-adjoint in L 2 (R 2 ), the operator H 0, is self-adjoint in L 2 (R), and hence the operator H 0 is self-adjoint in L 2 (R 3 ). Moreover, we have σ(H 0,⊥ ) = ∪ ∞ q=0 {2bq}, and every eigenvalue 2bq of H 0,⊥ has infinite multiplicity (see e.g. [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF]). Denote by p q the orthogonal projection onto Ker (H 0,⊥ -2bq), q ∈ Z + . Since σ(H

0, ) = [0, ∞), we have σ(H 0 ) = ∪ ∞ q=0 [2bq, ∞) = [0, ∞). Let now m ∈ Z, ̺ = (x 2 1 + x 2 2 ) 1/2 . Put H (m) 0,⊥ := - 1 ̺ ∂ ∂̺ ̺ ∂ ∂̺ + m ̺ - b̺ 2 2 -b.
The operator H (m) 0,⊥ is self-adjoint in L 2 (R + ; ̺d̺), and we have σ(H

(m) 0,⊥ ) = ∪ ∞ q=m -{2bq}
where m -= max{0, -m} (see e.g. [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF]). In contrast to the operator H 0,⊥ , every eigenvalue 2bq of H (m) 0,⊥ is simple. Denote by pq,m the orthogonal projection onto Ker (H

(m) 0,⊥ - 2bq), q ∈ Z + , q ≥ m -. Put (2.1) ϕ q,m (̺) := q! π(q + m)! b 2 m+1 ̺ m L (m) q b̺ 2 /2 e -b̺ 2 /4 , ̺ ∈ R + , q ∈ Z, q ≥ m -, where (2.2) L (m) q (s) := q l=m - (q + m)! (m + l)!(q -l)! (-s) l l! , s ∈ R,
are the generalized Laguerre polynomials. Then we have

H (m) 0,⊥ ϕ q,m = 2bqϕ q,m , ϕ q,m L 2 (R + ;̺d̺) = 1, and ϕ q,m = ϕ q,m (see e.g. [9, Section 9]). Moreover, pq,m = |ϕ q,m ϕ q,m |. Set H (m) 0 := H (m) 0,⊥ ⊗ I + Ĩ⊥ ⊗ H 0, where Ĩ⊥ is the identity operator in L 2 (R + ; ̺d̺). Evidently, σ(H (m) 0 ) = [2m -b, ∞).
Let (̺, φ, x 3 ) be the cylindrical coordinates in R 3 . The operator H (m) 0 , m ∈ Z, is unitarily equivalent to the restriction of H 0 onto Ker (Lm) where

L := -i x 1 ∂ ∂x 2 -x 2 ∂ ∂x 1 = -i ∂ ∂φ
is the x 3 -component of the angular-momentum operator, which commutes with H 0 . Moreover, the operator H 0 is unitarily equivalent to the orthogonal sum ⊕ m∈Z H (m) 0 . More precisely, if we pass to cylindrical coordinates, and decompose u ∈ Dom(H 0 ) into a Fourier series with respect to φ, i.e. if we write

u(̺ cos φ, ̺ sin φ, x 3 ) = m∈Z e imφ u m (̺, x 3 ), we have (H 0 u)(̺ cos φ, ̺ sin φ, x 3 ) = m∈Z e imφ (H (m) 0 u m )(̺, x 3 ).

2.2.

In this subsection we perturb the operators H (m) 0

and H 0 by a scalar potential v 0 which depends only on the variable x 3 . Let v 0 : R → R be a measurable function. Throughout the paper we assume that the multiplier by v 0 is H 0, -compact, which is ensured, for instance, by

v 0 ∈ L 2 (R) + L ∞ ε (R). Set H := H 0, + v 0 .
Then we have σ ess (H ) = σ ess (H 0, ) = [0, ∞). For simplicity, throughout the article we suppose also that (2.3) inf σ(H ) > -2b.

Evidently, (2.3) holds true if the negative part v 0,-of the function v 0 is bounded, and we have v 0,-L ∞ (R) < 2b. Assume now that the discrete spectrum of H is not empty; this would follow, for example, from the additional conditions v 0 ∈ L 1 (R) and R v 0 (x)dx < 0 (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF]Theorem XIII.110]). Occasionally, we will impose also the assumption that the discrete spectrum of H consists of a unique eigenvalue; this would be implied, for instance, by the inequality R |x|v 0,-(x)dx < 1 (see e.g. [5, Chaper II, Theorem 5.1]).

Let λ be a discrete eigenvalue of the operator H which necessarily is simple. Then λ ∈ (-2b, 0) by (2.3). Let ψ be an eigenfunction satisfying

(2.4) H ψ = λψ, ψ L 2 (R) = 1, ψ = ψ.
Denote by p = p (λ) the spectral projection onto Ker(Hλ). We have p = |ψ ψ|. Suppose now that v 0 satisfies (2.5)

|v 0 (x)| = O x -m 0 , x ∈ R, m 0 > 1,
where x := (1+|x| 2 ) 1 2 . Then the multiplier by v 0 is a relatively trace-class perturbation of H 0, , and by the Birman-Kuroda theorem (see e.g. [33, Theorem XI.9]) we have

σ ac (H ) = σ ac (H 0, ) = [0, ∞).
Moreover, by the Kato theorem (see e.g. [START_REF] Reed | Methods of Modern Mathematical Physics. IV. Analysis of Operators[END_REF]Theorem XIII.58]) the operator H has no strictly positive eigenvalues. In fact, for all E > 0 and s > 1/2 the operator-norm limit (2.6)

x -s (H -E) -1 x -s := lim δ↓0

x -s (H -Eiδ) -1 x -s , exists in L(L 2 (R)), and for each compact subset J of R + = (0, ∞) and each s > 1/2 there exists a constant C J,s such that for each E ∈ J we have

(2.7) x -s (H -E) -1 x -s ≤ C J,s
(see [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF]). Suppose again that (2.5) holds true, and let us consider the differential equation

(2.8) -y ′′ + v 0 y = k 2 y, k ∈ R.
It is well-known that (2.8) admits the so-called Jost solutions y 1 (x; k) and y 2 (x; k) which obey

y 1 (x; k) = e ikx (1 + o(1)), x → ∞, y 2 (x; k) = e -ikx (1 + o(1)), x → -∞,
uniformly with respect to k ∈ R (see e.g. [5, Chapter II, Section 6] or [START_REF] Tang | Potential Scattering on the Real Line[END_REF]). The pairs {y l (•; k), y l (•; -k)}, k ∈ R, l = 1, 2, form fundamental sets of solutions of (2.8). Define the transition coefficient T (k) and the reflection coefficient R(k), k ∈ R, k = 0, by

y 2 (x; k) = T (k)y 1 (x; -k) + R(k)y 1 (x; k), x ∈ R.
It is well known that T (k) = 0, k ∈ R \ {0}. On the other hand, the Wronskian of the solutions y 1 (•; k) and y 2 (•; k) is equal to -2ikT (k), and hence these solutions are linearly independent for k ∈ R \ {0}. For E > 0 set

Ψ l (x; E) := 1 4π √ ET ( √ E) y l (x; √ E), l = 1, 2. Evidently, Ψ l (•; E) ∈ C 1 (R) ∩ L ∞ (R), E > 0, l = 1, 2.
Moreover, the real and the imaginary part of both functions Ψ l (•; E), l = 1, 2 with E > 0 do not vanish identically. Further, Im x -s (H -E) -1 x -s with E > 0 and s > 1/2 is a rank-two operator with an integral kernel

K(x, x ′ ) = π l=1,2 x -s Ψ l (x; E)Ψ l (x ′ ; E) x ′ -s , x, x ′ ∈ R.

2.3.

Fix now m ∈ Z, and set

H (m) := H (m) 0 + Ĩ⊥ ⊗ v 0 .
Since v 0 is H 0, -compact and the spectrum of

H (m) 0,⊥ is discrete, the operator Ĩ⊥ ⊗ v 0 is H (m)
0 -compact. Therefore, the operator H (m) is well-defined on Dom(H (m) 0 ), and we have

σ ess (H (m) ) = σ ess (H (m) 0 ) = [2bm -, ∞). Further, if λ is a discrete eigenvalue of H , then 2bq + λ is a simple eigenvalue of H (m)
for each integer q ≥ m -. If q = m -, then this eigenvalue is isolated, but if q > m -, then due to (2.3), it is embedded in the essential spectrum of H (m) . Moreover, (2.9) H (m) Φ q,m = (2bq + λ)Φ q,m , q ≥ m -, Φ q,m = ϕ q,m ⊗ ψ, ϕ q,m being defined in (2.1), and ψ in (2.4). Set

(2.10) P q,m := pq,m ⊗ p .

Then we have P q,m = |Φ q,m Φ q,m |. Finally, introduce the operator

H := H 0 + I ⊥ ⊗ v 0 .
Even though the operator I ⊥ ⊗ v 0 is not H 0 -compact (unless v 0 = 0), it is H 0 -bounded with zero relative bound so that the operator H is well-defined on Dom(H 0 ). Evidently, the operator H is unitarily equivalent to the orthogonal sum ⊕ m∈Z H (m) . Up to the additive constant b, the operator H is the Hamiltonian of a quantum non-relativistic spinless particle in an electromagnetic field (E, B) with parallel electric component E = -(0, 0, v ′ 0 (x 3 )), and magnetic component B = (0, 0, b). Note that if λ is a discrete eigenvalue of H , then 2bq + λ, q ∈ Z + is an eigenvalue of infinite multiplicity of H. If q = 0, this eigenvalue is isolated, and if q ≥ 1, it lies on the interval [0, ∞) which constitutes a part of the essential spectrum of H. Moreover, if (2.5) holds, then σ ac (H) = [0, ∞), so that in this case 2bq + λ, q ∈ Z + , is embedded in the absolutely continuous spectrum of H.

2.4.

In this subsection we introduce appropriate perturbations of the operators H and H (m) , m ∈ Z + . Let V : R 3 → R be a measurable function. Assume that V is H 0 -bounded with zero relative bound. By the diamagnetic inequality (see e.g. [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF]) this would follow, for instance, from V ∈ L 2 (R 3 ) + L ∞ (R 3 ). On Dom(H) = Dom(H 0 ) define the operator H + κV , κ ∈ R. Remark: We impose the condition that the relative H 0 -bound is zero just for the sake of simplicity. If V is H 0 -bounded with arbitrary finite relative bound, then we could again define H + κV but only for sufficiently small |κ|. Occasionally, we will impose the more restrictive assumption that

V is H 0 -compact; this would follow from V ∈ L 2 (R 3 ) + L ∞ ε (R 3 ). In particular, V is H 0 -compact if it satisfies the estimate (2.11) |V (x)| = O X ⊥ -m ⊥ x 3 -m 3 , x = (X ⊥ , x 3 ), m ⊥ > 0, m 3 > 0.
Further, assume that V is axisymmetric, i.e. V depends only on the variables (̺, x 3 ). Fix m ∈ Z and assume that the multiplier by V is H (m) 0 -bounded with zero relative bound. Then the operator

H (m) + V is well defined on Dom(H (m) ) = Dom(H (m) 0 ). Define the operator H (m) + κV , κ ∈ R. For z ∈ C + := {ζ ∈ C | Im ζ > 0}, m ∈ Z, q ≥ m -, introduce the quantity F q,m (z) := (H (m) -z) -1 (I -P q,m )V Φ q,m , V Φ q,m
where •, • denotes the scalar product in L 2 (R + × R; ̺d̺dx 3 ), which we define to be linear with respect to the first factor. If λ is a discrete eigenvalue of H we will say that the Fermi Golden Rule F q,m,λ is valid if the limit (2.12)

F q,m (2bq + λ) = lim δ↓0 F q,m (2bq + λ + iδ),
exists and is finite, and

(2.13) Im F q,m (2bq + λ) > 0.

Resonances via dilation analyticity

3.1. In this subsection we will perturb H (m) 0 by an axisymmetric potential V (̺, x 3 ) so that the simple eigenvalue 2bq + λ of H (m) 0 becomes a resonance of the perturbed operator. In order to use complex scaling, we impose an analyticity assumption. We assume that the potential v 0 extends to an analytic function in the sector

S θ 0 = {z ∈ C | |Argz| ≤ θ 0 , or |z| ≤ r 0 }
with θ 0 ∈ (0, π/2), which satisfies (2.5). As already used in similar situations (see e.g. [START_REF] Avron | Schrödinger operators with magnetic fields[END_REF], [START_REF] Wang | Barrier resonances in strong magnetic fields[END_REF]) we introduce complex deformation in the longitudinal variable, (U(θ)f )(̺,

x 3 ) = e θ/2 f (̺, e θ x 3 ), f ∈ L 2 (R + × R; ̺d̺dx 3 ), θ ∈ R. For θ ∈ R we have H (m) (θ) = U(θ)H (m) U -1 (θ) = H (m) 0 ⊗ I + I ⊗ H (θ), with H (θ) = -e -2θ d 2 dx 2 3 + v 0,θ (x 3
), and v 0,θ (x 3 ) := v 0 (e θ x 3 ). By assumption, the family of operators {H (θ), |Im θ| < θ 0 }, form a type (A) analytic family of m-sectorial operators in the sense of Kato (see for instance [START_REF] Hislop | Introduction to spectral theory. With applications to Schrödinger operators[END_REF]Section 15.4], [START_REF] Aguilar | A class of analytic perturbations for one-body Schrödinger Hamiltonians[END_REF]). Then the discrete spectrum of H 0, (θ) is independent of θ and we have

σ(H (m) (θ)) = q≥m - {2bq + σ(H (θ))}, σ(H (θ)) = e -2θ R + ∪ σ disc (H ) ∪ {z 1 , z 2 , ....}
where σ disc (H ) denotes the discrete spectrum of H , and z 1 , z 2 , .... are (complex) eigenvalues of H (θ) in {0 > Arg z > -2Im θ}, Im θ > 0. In the sequel, we assume that σ disc (H ) = {λ}. Further, we assume that V is axisymmetric, and admits an analytic extension with respect to

x 3 ∈ S θ 0 , which is H (m) 0 -compact (see e.g. [34, Chapter XII]). Let V θ (̺, x 3 ) := V (̺, e θ x 3 ).
Then the family of operators {H (m) (θ) + κV θ , |Im θ| < θ 0 , |κ| ≤ 1}, form also an analytic family of type (A) for sufficiently small κ.

By definition, the resonances of

H (m) + κV in S m -(θ) := q≥m - {z ∈ C; 2bq < Rez < 2b(q + 1), -2Imθ < Arg(z -2bq) ≤ 0}
are the eigenvalues of H (m) (θ) + κV θ , Imθ > 0. For V axisymmetric and H 0 -compact, we define the set of the resonances Res(H + κV, S 0 (θ)) of the operator H + κV in S 0 (θ) by

Res(H + κV, S 0 (θ)) := m∈Z {eigenvalues of H (m) (θ) + κV θ } ∩ S 0 (θ).
In other words, the set of resonances of H + κV is the union with respect to m ∈ Z of the resonances of H (m) + κV . This definition is correct since the restriction of H + κV onto Ker(Lm) is unitarily equivalent to H (m) + κV . Moreover using a standard deformation argument (see, for instance, [START_REF] Hislop | Introduction to spectral theory. With applications to Schrödinger operators[END_REF]Chapter 16]), we can prove that these resonances coincide with singularities of the function z

→ (H + κV -z) -1 f, f , for f in a dense subset of L 2 (R 3 ). Theorem 3.1. Fix m ∈ Z, q > m -. Assume that:
• v 0 admits an analytic extension in S θ 0 which satisfies (2.5);

• inequality (2.3) holds true, and H has a unique discrete eigenvalue λ;

• V is axisymmetric, and admits an analytic extension with respect to x 3 in S θ 0 which is H (m) 0 -compact. Then for sufficiently small |κ|, the operator H (m) + κV has a resonance w q,m (κ) which obeys the asymptotics

(3.1) w q,m (κ) = 2bq + λ + κ V Φ q,m , Φ q,m -κ 2 F q,m (2bq + λ) + O q,m,V (κ 3 ), κ → 0,
the eigenfunction Φ q,m being defined in (2.9), and the quantity F q,m (2bq+λ) being defined in (2.12).

Proof. Fix θ such that θ 0 > Im θ ≥ 0 and assume that z ∈ C is in the resolvent set of the operator

H (m) (θ) + κV θ . Put R (m) κ,θ (z) := (H (m) (θ) + κV θ -z) -1 .
By the resolvent identity, we have

(3.2) R (m) κ,θ (z) = R (m) 0,θ (z)-κ R (m) 0,θ (z)V θ R (m) 0,θ (z)+κ 2 R (m) 0,θ (z)V θ R (m) 0,θ (z)V θ R (m)
0,θ (z)+O(κ 3 ), as κ → 0, uniformly with respect to z in a compact subset of the resolvent sets of H (m) (θ) + κV and H (m) (θ). Now note that the simple embedded eigenvalue 2bq + λ of H (m) is a simple isolated eigenvalue of H (m) (θ). According to the Kato perturbation theory (see [22, Section VIII.2]), for sufficiently small κ there exists a simple eigenvalue w q,m (κ) of H (m) (θ)+κV θ such that lim κ→0 w q,m (κ) = w q,m (0) = 2bq + λ. For |κ| sufficiently small, define the eigenprojector

(3.3) P κ (θ) = P κ,q,m (θ) := -1 2iπ Γ R (m) κ,θ (z)dz
where Γ is a small positively oriented circle centered at 2bq + λ. Evidently, for u ∈ Ran P κ (θ) we have (H (m) (θ) + κV θ )u = w q,m (κ)u; in particular, if u ∈ Ran P 0 (θ), then H (m) (θ)u = (2bq + λ)u. Since w q,m (κ) is a simple eigenvalue, we have

(3.4) w q,m (κ) = Tr -1 2iπ Γ zR (m)
κ,θ (z)dz for Γ and κ as above. Inserting (3.2) into (3.4), we get

w q,m (κ) = (3.5) 2bq + λ + κ Tr(P 0 (θ) V θ P 0 (θ)) - κ 2 2iπ Tr Γ zR (m) 0,θ (z) V θ R (m) 0,θ (z) V θ R (m) 0,θ (z) dz + O(κ 3 )
as κ → 0. Next, we have

(3.6) Tr(P 0 (θ) V θ P 0 (θ)) = Tr(P q,m V P q,m ) = V Φ q,m , Φ q,m L 2 (R + ×R;̺d̺dx 3 ) ,
the orthogonal projection P q,m being defined in (2.10). For θ ∈ R the relation is obvious since the operators P 0,q,m (θ) V θ P 0,q,m (θ) and P q,m V P q,m are unitarily equivalent. For general complex θ identity (3.6) follows from the fact the function θ → Tr(P 0,q,m (θ)

V θ P 0,q,m (θ)) is analytic. Set Q 0 (θ) := I -P 0 (θ), H(m) (θ) := H (m) (θ)Q 0 (θ).
By the cyclicity of the trace, we have

(3.7) Tr Γ zR (m) 0,θ (z) V θ R (m) 0,θ (z) V θ R (m) 0,θ (z)dz = T 1 + T 2 + T 3 + T 4
where

T 1 := Γ z(2bq + λ -z) -3 Tr (P 0 (θ) V θ P 0 (θ)V θ P 0 (θ)) dz, T 2 := Γ z(2bq + λ -z) -2 Tr P 0 (θ) V θ ( H(m) (θ) -z) -1 Q 0 (θ) V θ P 0 (θ) dz, T 3 := Tr Γ z( H(m) (θ) -z) -1 Q 0 (θ)V θ ( H(m) (θ) -z) -1 Q 0 (θ)V θ ( H(m) (θ) -z) -1 Q 0 (θ)dz , T 4 := Γ z (2bq + λ -z) -1 Tr P 0 (θ) V θ ( H(m) (θ) -z) -2 Q 0 (θ) V θ P 0 (θ) dz.
Since Γ z(2bq + λz) -3 dz = 0, and the function z → ( H(m) (θ)z) -1 is analytic inside Γ, we have

(3.8) T 1 = T 3 = 0.
Further, using the identity

(2bq + λ -z) -2 P 0 (θ) V θ ( H(m) (θ) -z) -1 Q 0 (θ) V θ P 0 (θ)+ (2bq + λ -z) -1 P 0 (θ) V θ ( H(m) (θ) -z) -2 Q 0 (θ) V θ P 0 (θ) = ∂ ∂z (2bq + λ -z) -1 P 0 (θ) V θ ( H(m) (θ) -z) -1 Q 0 (θ) V θ P 0 (θ) ,
integrating by parts, and applying the Cauchy theorem, we obtain (3.9)

T 2 + T 4 = 2iπ Tr P 0 (θ) V θ ( H(m) (θ) -2bq -λ) -1 (I -P 0 (θ)) V θ P 0 (θ) .
Arguing as in the proof of (3.6), we get (3.10)

Tr P 0 (θ) V θ ( H(m) (θ) -2bq -λ -iδ) -1 Q 0 (θ) V θ P 0 (θ) = F q,m (2bq + λ + iδ), δ > 0.
For θ fixed such that Im θ > 0, the point 2bq + λ is not in the spectrum of H(m) (θ).

Taking the limit δ ↓ 0 in (3.10), we find that (3.9) implies

(3.11) T 2 + T 4 = 2iπ F q,m (2bq + λ).
Putting together (3.5) -(3.8) and (3.11), we deduce (3.1).

Remarks: (i) We will see in Section 4 that generically Im F q,m (2bq +λ) > 0 for all m ∈ Z, and q > m -.

(ii) Taking into account the above remark, we find that Theorem 3.1 implies that generically near 2bq+λ, q ≥ 1, there are infinitely many resonances of H +κV with sufficiently small κ, namely the resonances of the operators H (m) + κV with m > -q.

3.2.

In this subsection we consider the dynamical aspect of resonances. We prove the following proposition which will be extended to non-analytic perturbations in Section 4.

Proposition 3.1. Under the assumptions of Theorem 3.1 there exists a function g ∈ C ∞ 0 (R; R) such that g = 1 near 2bq + λ, and (3.12)

e -i(H (m) +κV )t g(H (m) + κV )Φ q,m , Φ q,m = a(κ)e -iwq,m(κ)t + b(κ, t), t ≥ 0, with a and b satisfying the asymptotic estimates

|a(κ) -1| = O(κ 2 ), |b(κ, t)| = O(κ 2 (1 + t) -n ), ∀n ∈ Z + , as κ → 0 uniformly with respect to t ≥ 0.
In order to prove the proposition, we will need the following

Lemma 3.1. Set Q κ (θ) := I -P κ (θ), R(m) κ,θ (z) := (H (m) (θ) + κV θ )Q κ (θ) -z -1 Q κ (θ),
the projection P κ (θ) being defined in (3.3). Then for |κ| small enough, there exists a finite-rank operator

F (m)
κ,θ , uniformly bounded with respect to κ, such that

(3.13) P 0 (θ) = P κ (θ) + κ R(m) κ,θ (w q,m (κ))V θ P κ (θ) + P κ (θ)V θ R(m) κ,θ (w q,m (κ)) + κ 2 F (m)
κ,θ . Proof. By the resolvent identity, we have

(3.14) R (m) 0,θ (ν) = R (m) κ,θ (ν) + κ R (m) κ,θ (ν)V θ R (m) κ,θ (ν) + κ 2 R (m) κ,θ (ν)V θ R (m) κ,θ (ν)V θ R (m) 0,θ (ν).
Moreover by definition of R(m) κ,θ and of H(m) := H (m) Q 0 , we have:

(3.15) R (m) κ,θ (ν) = (w q,m (κ) -ν) -1 P κ (θ) + R(m) κ,θ (ν), R (m) 0,θ (ν) = (2bq + λ -ν) -1 P 0 (θ) + ( H(m) (θ) -ν) -1 Q 0 (θ), where ν → R(m) κ,θ (ν) and ν → ( H(m) (θ) -ν) -1 Q 0 (θ) are analytic near 2bq + λ.
Then, from the Cauchy formula, the integration of (3.14) on a small positively oriented circle centered at 2bq + λ, yields (3.13) with F (m) κ,θ a linear combination of finite-rank operators of the form P 1 V θ P 2 V θ P 3 , where {P 0 (θ), P κ (θ)} ∩ {P j , j = 1, 2, 3} = ∅, and

P j ∈ {P 0 (θ), P κ (θ), R(m) κ,θ (ν), ( H(m) (θ) -ν) -1 Q 0 (θ), with ν = w q,m (
κ) or ν = 2bq + λ}. Since these operators are uniformly bounded in κ with |κ| small enough, F (m) κ,θ is a finite-rank operator which also is uniformly bounded in κ with |κ| small enough.

Proof of Proposition 3.1: Pick at first any g ∈ C ∞ 0 (R; R) such that g = 1 near 2bq + λ. We have e -i(H (m) +κV )t g(H (m) + κV )Φ q,m , Φ q,m = Tr (e -i(H (m) +κV )t g(H (m) + κV )P q,m ).

By the Helffer-Sjöstrand formula, (3.16) e -i(H (m) +κV )t g(H (m) +κV )

P q,m = 1 π R 2 ∂g ∂ z (z) e -izt (H (m) +κV -z) -1 P q,m dxdy
where z = x + iy, z = xiy, g is a compactly supported, quasi-analytic extension of g, and the convergence of the integral is understood in the operator-norm sense (see e.g. [START_REF] Dimassi | Spectral Asymptotics in the Semi-Classical Limit[END_REF]Chapter 8]).

Consider the functions σ ± (z) := Tr ((H (m) + κVz) -1 P q,m ), ±Imz > 0.

Following the arguments of the previous subsection, we find that

(3.17) σ + (z) = Tr (R (m)
κ,θ (z) P 0,q,m (θ)), Im z > 0, θ 0 > Im θ > 0. Inserting (3.13) into (3.17), and using the cyclicity of the trace, and the elementary identities

P κ (θ) R (m) κ,θ (z) R(m) κ,θ (w q,m (κ)) = 0 = R(m) κ,θ (w q,m (κ)) P κ (θ) R (m) κ,θ (z), we get σ + (z) = Tr (R (m) κ,θ (z) P κ (θ)) + κ 2 Tr (R (m) κ,θ (z) F (m)
κ,θ ). Applying (3.15), we obtain

(3.18) σ + (z) = 1 + κ 2 r(κ) (w q,m (κ) -z) -1 + κ 2 G + (κ, z),
where r(κ) := Tr (P κ (θ) F

κ,θ ), and G + (κ, z) is analytic near 2bq + λ and uniformly bounded with respect to |κ| small enough. Similarly, (3.19) σ

-(z) = 1 + κ 2 r(κ) (w q,m (κ) -z) -1 + κ 2 G -(κ, z),
where G -(κ, z) is analytic near 2bq + λ and uniformly bounded with respect to |κ| small enough. Now, assume that the support of g is such that we can choose g supported on a neighborhood of 2bq + λ where the functions z → G ± (κ, z) are holomorphic. Combining (3.16) with the Green formula, we get (3.20) Tr (e -i(H (m) +κV )t g(H (m) + κV

) P q,m ) = 1 2iπ R g(µ) e -iµt (σ + (µ) -σ -(µ))dµ.
Making use of (3.18) -(3.19), we get

1 2iπ R g(µ) e -iµt (σ + (µ) -σ -(µ))dµ = κ 2 2iπ R g(µ) e -iµt (G + (κ, µ) -G -(κ, µ))dµ + 1 + κ 2 r(κ) 2iπ R g(µ) e -iµt (w q,m (κ) -µ) -1 dµ - 1 + κ 2 r(κ) 2iπ R g(µ) e -iµt (w q,m (κ) -µ) -1 dµ. Pick ε > 0 so small that g(µ) = 1 for µ ∈ [2bq + λ -2ε, 2bq + λ + 2ε]. Set C ε := (-∞, 2bq + λ -ε] ∪ {2bq + λ + εe it , t ∈ [-π, 0]} ∪ [2bq + λ + ε, +∞), g(µ) := 1, µ ∈ C ε \ R.
Taking into account (3.20), bearing in mind that Im w q,m (κ) < 0, and applying the Cauchy theorem, we easily find that (3.21) Tr (e -i(H (m) +κV )t g(H (m) +κV ) P q,m ) = (1+κ 2 r(κ))e -iwq,m(κ)t +κ 2 j=1,2,3

I j (t; κ)
where

I 1 (t; κ) := 1 2iπ R g(µ) e -iµt (G + (κ, µ) -G -(κ, µ))dµ, I 2 (t; κ) := 1 2iπ Cε g(µ) e -iµt (r(κ)(w q,m (κ) -µ) -1 -r(κ)(w q,m (κ) -µ) -1 )dµ, I 3 (t; κ) := - Im w q,m (κ) κ 2 π Cε g(µ) e -iµt (w q,m (κ) -µ) -1 (w q,m (κ) -µ) -1 dµ.
Integrating by parts, we find that

(3.22) |I j (t; κ)| = O((1 + t) -n ), t > 0, j = 1, 2, 3, ∀n ∈ Z + ,
uniformly with respect to κ, provided that |κ| is small enough; in the estimate of I 3 (t; κ) we have taken into account that by Theorem 3.1 we have |Im(w q,m (κ))| = O(κ 2 ). Putting together (3.21) and (3.22), we get (3.12).

Mourre estimates and dynamical resonances

In this section we obtain Mourre estimates for the operator H (m) and apply them combined with a recent result of Cattaneo, Graf, and Hunziker (see [START_REF] Cattaneo | A general resonance theory based on Mourre's inequality[END_REF]) in order to investigate the dynamics of the resonance states of the operator H (m) without analytic assumptions.

Let v

0 : R → R. Set (4.1) v j (x 3 ) := x j 3 v (j) 0 (x 3 ), j ∈ Z + , provided that the corresponding derivative v (j) 0 of v 0 is well-defined. Let A := - i 2 x 3 d dx 3 + d dx 3 x 3
be the self-adjoint operator defined initially on C ∞ 0 (R) and then closed in L 2 (R). Set A := Ĩ⊥ ⊗ A. Let T be an operator self-adjoint in L 2 (R + × R; ̺d̺dx 3 ) such that e isA D(T ) ⊆ D(T ), s ∈ R. Define the commutator [T, iA] in the sense of [START_REF] Jensen | Multiple commutator estimates and resolvent smoothness in quantum scattering theory Ann[END_REF] and [START_REF] Cattaneo | A general resonance theory based on Mourre's inequality[END_REF], and set ad A (T ), iA], k ≥ 1, provided that the higher order commutators are well-defined. Evidently, for each m ∈ Z we have

(4.2) i k ad (k) A (H (m) ) = 2 k Ĩ⊥ ⊗ H 0, + k j=1 c k,j v j , k ∈ Z + ,
with some constants c k,j independent of m; in particular, c k,k = (-1) k . Therefore the H 0, -boundedness of the multipliers v j , j = 1, . . . , k, guarantees the H (m) -boundedness of all the operators ad (j)

A (H (m) ) , j = 1, . . . , k. Let J ⊂ R be a Borel set, and T be a self-adjoint operator. Denote by P J (T ) the spectral projection of the operator T associated with J.

Proposition 4.1. Fix m ∈ Z. Let λ ∈ (-2b, 0), q ∈ Z, q > m -. Put (4.3) J = (2bq + λ -δ, 2bq + λ + δ)
where δ > 0, δ < -λ/2, and δ < (2b + λ)/2. Assume that the operators v j (H 0, + 1) -1 , j = 0, 1, are compact. Then there exist a positive constant C > 0 and a compact operator K such that (4.4)

P J (H (m) )[H (m) , iA]P J (H (m) ) ≥ CP J (H (m) ) + K. Proof. Let χ ∈ C ∞ 0 (R; R) be such that supp χ = [2bq + λ -2δ, 2bq + λ + 2δ], χ(t) ∈ [0, 1], ∀t ∈ R, χ(t) = 1, ∀t ∈ J.
In order to prove (4.4), it suffices to show that

(4.5) χ(H (m) )[H (m) , iA]χ(H (m) ) ≥ Cχ(H (m) ) 2 + K
with a compact operator K. Indeed, if inequality (4.5) holds true, we can multiply it from the left and from the right by P J (H (m) ) obtaining thus (4.4) with

K = P J (H (m) ) KP J (H (m) ). Next (4.2) yields [H (m) , iA] = 2 Ĩ⊥ ⊗ H 0, -v 1 . Therefore, χ(H (m) )[H (m) , iA]χ(H (m) ) = 2χ(H (m) ) Ĩ⊥ ⊗ H 0, χ(H (m) ) -χ(H (m) )v 1 χ(H (m) ) = (4.6) 2χ(H (m) 0 ) Ĩ⊥ ⊗ H 0, χ(H (m) 0 ) + 2K 1 -K 2 where K 1 := χ(H (m) ) Ĩ⊥ ⊗ H 0, χ(H (m) ) -χ(H (m) 0 ) Ĩ⊥ ⊗ H 0, χ(H (m) 0 ), K 2 := χ(H (m) )v 1 χ(H (m) ). Since the operator v 1 (H (m) 0 + 1) -1 is compact, the operators v 1 (H (m) + 1) -1 , v 1 χ(H (m)
), and K 2 are compact as well. Let us show that K 1 is also compact. We have

K 1 = (χ(H (m) ) -χ(H (m) 0 )) Ĩ⊥ ⊗ H 0, χ(H (m) )+ (4.7) χ(H (m) 0 ) Ĩ⊥ ⊗ H 0, (χ(H (m) ) -χ(H (m) 0
)). By the Helffer-Sjöstrand formula and the resolvent identity,

χ(H (m) ) -χ(H (m) 0 ) = - 1 π R 2 ∂ χ ∂ z (z)(H (m) -z) -1 v 0 (H (m) 0 -z) -1 dxdy.
Since the support of χ is compact in R 2 , and the operator

∂ χ ∂ z (H (m) -z) -1 v 0 (H (m) 0 -z) -1
is compact for every (x, y) ∈ R 2 with y = 0, and is uniformly norm-bounded for every (x, y) ∈ R 2 , we find that the operator χ(H (m) )χ(H (m) 0 ) is compact. On the other hand, it is easy to see that the operators

Ĩ⊥ ⊗ H 0, χ(H (m) ) = Ĩ⊥ ⊗ H 0, (H (m) + 1) -1 (H (m) + 1)χ(H (m) ) = Ĩ⊥ ⊗ H 0, (H (m) 0 + 1) -1 (I -v 0 (H (m) + 1) -1 )(H (m) + 1)χ(H (m) )
and χ(H

(m) 0 ) Ĩ⊥ ⊗ H 0, = χ(H (m) 0 )(H (m) 0 + 1)(H (m) 0 + 1) -1 Ĩ⊥ ⊗ H 0,
are bounded. Taking into account (4.7), and bearing in mind the compactness of the operator χ(H (m) )-χ(H (m) 0 ), and the boundedness of the operators Ĩ⊥ ⊗ H 0, χ(H (m) ) and χ(H (m) 0 ) Ĩ⊥ ⊗ H 0, , we conclude that the operator K 1 is compact. Further, since δ < -λ/2, and hence 2bj > 2bq + λ + 2δ for all j ≥ q, we have

χ(H 0, + 2bj) = 0, j ≥ q.
Therefore,

χ(H (m) 0 ) = ∞ j=m - pj,m ⊗ χ(H 0, + 2bj) = q-1 j=m - pj,m ⊗ χ(H 0, + 2bj),

and (4.8)

χ(H

(m) 0 ) Ĩ⊥ ⊗ H 0, χ(H (m) 0 ) = q-1 j=m - pj,m ⊗ χ(H 0, + 2bj) 2 H 0, .
By the spectral theorem,

q-1 j=m - pj,m ⊗ χ(H 0, + 2bj) 2 H 0, ≥ q-1 j=m - (2b(q -j) + λ -2δ)p j,m ⊗ χ(H 0, + 2bj) 2 ≥ (4.9) (2b + λ -2δ) q-1 j=m - pj,m ⊗ χ(H 0, + 2bj) 2 = C 1 χ(H (m) 0 ) 2
with C 1 := 2b + λ -2δ > 0. Combining (4.6), (4.8), and (4.9), we get

(4.10) χ(H (m) )[H (m) , iA]χ(H (m) ) ≥ 2C 1 χ(H (m) ) 2 + 2C 1 K 3 + 2K 1 -K 2 where K 3 := χ(H (m) 0 ) 2 -χ(H (m)
) 2 is a compact operator by the Helffer-Sjöstrand formula. Now we find that (4.10) is equivalent to (4.5) with C = 2C 1 andK

= 2C 1 K 3 + 2K 1 -K 2 .
Remark: Mourre estimates for various magnetic quantum Hamiltonians can be found in [START_REF] Gérard | Multiparticle Quantum Scattering in Constant Magnetic Fields[END_REF]Chapter 3].

4.2.

By analogy with (4.1) set

V j (̺, x 3 ) = x j 3 ∂ j V (̺, x 3 ) ∂x j 3 , j ∈ Z + .
We have

i k ad (k) A (V ) = k j=1 c k,j V j
with the same constants c k,j as in (4.2). We will say that the condition O ν , ν ∈ Z + , holds true if the multipliers by v j , j = 0, 1, are H 0, -compact, and the multipliers by v j , j ≤ ν, are H 0, -bounded. Also, for a fixed m ∈ Z we will say that the condition C ν,m , ν ∈ Z + , holds true if the condition O ν is valid, the multiplier by is H (m) 0 -bounded with zero relative bound, and the multipliers by V j , j = 1, . . . , ν, are H (m) 0 -bounded. By Proposition 4.1 and [11, Lemma 3.1], the validity of condition C ν,m with ν ≥ 5 and a given m ∈ Z guarantees the existence of a finite limit F q,m (2bq + λ) with q > m -in (2.12), provided that (2.3) holds true, and λ is a discrete eigenvalue of H 0, + v 0 . Combining the results of Proposition 4.1 and [11, Theorem 1.2], we obtain the following Theorem 4.1. Fix m ∈ Z, n ∈ Z + . Assume that:

• the condition C ν,m holds with ν ≥ n + 5;

• inequality (2.3) holds true, and λ is a discrete eigenvalue of H ;

• inequality (2.13) holds true, and hence the Fermi Golden Rule F q,m,λ is valid.

Then there exists a function g ∈ C ∞ 0 (R; R) such that supp g = J (see (4.3)), g = 1 near 2bq + λ, and

(4.11)
e -i(H (m) +κV )t g(H (m) + κV )Φ q,m , Φ q,m = a(κ)e -iλq,m(κ)t + b(κ, t), t ≥ 0, where (4.12) λ q,m (κ

) = 2bq + λ + κ V Φ q,m , Φ q,m -κ 2 F q,m (2bq + λ) + o q,m,V (κ 2 ), κ → 0.
In particular, we have Im λ q,m (κ) < 0 for |κ| small enough. Moreover, a and b satisfy the asymptotic estimates

|a(κ) -1| = O(κ 2 ), |b(κ, t)| = O(κ 2 | ln |κ||(1 + t) -n ), |b(κ, t)| = O(κ 2 (1 + t) -n+1 ),
as κ → 0 uniformly with respect to t ≥ 0.

We will say that the condition C ν , ν ∈ Z + , holds true if the condition O ν is valid, the multiplier by V is H 0 -bounded with zero relative bound, and the multipliers by V j , j = 1, . . . , ν, are H 0 -bounded.

For m ∈ Z and q ≥ m -denote by Φq,m : R 3 → C the function written in cylindrical coordinates (̺, φ, x 3 ) as Φq,m (̺, φ, x 3 ) = (2π) -1 2 e imφ Φ q,m (̺, x 3 ).

Corollary 4.1. Fix n ∈ Z + . Assume that:

• the condition C ν holds with ν ≥ n + 5;

• inequality (2.3) is fulfilled, and λ is a discrete eigenvalue of H 0, + v 0 ;

• for each m ∈ Z, q > m -, inequality (2.13) holds true, and hence the Fermi Golden Rule F q,m,λ is valid.

Then for every fixed q ∈ Z + , and each m ∈ {-q + 1, . . . , 0} ∪ N with N := {1, 2, . . .}, we have

e -i(H+κV )t g(H + κV ) Φq,m , Φq,m L 2 (R 3 ) = a(κ)e -iλq,m(κ)t + b(κ, t), t ≥ 0,
where g, λ q,m (κ), a, and b are the same as in Theorem 4.1.

Remarks: (i) If q ≥ 1, then Corollary 4.1 tells us that typically the eigenvalue 2bq + λ of the operator H, which has an infinite multiplicity, generates under the perturbation κV infinitely many resonances with non-zero imaginary part. Note however that 2bq + λ is a discrete simple eigenvalue of the operator H (-q) , and therefore the operator H (-q) + κV has a simple discrete eigenvalue provided that |κ| is small enough. Generically, this eigenvalue is an embedded eigenvalue for the operator H + κV .

(ii) If q = 0, then λ is an isolated eigenvalue of infinite multiplicity for H. By Theorem 6.1 below, in this case there exists an infinite series of discrete eigenvalues of the operator H + V which accumulate at λ, provided that the perturbation V has a definite sign.

Sufficient conditions for the validity of the Fermi Golden Rule

In this section we describe certain classes of perturbations V compatible with the hypotheses of Theorems 3.1 -4.1, for which the Fermi Golden rule F q,m,λ is valid for every m ∈ Z and q > m -. The results included are of two different types. Those of Subsection 5.1 are less general but they offer a constructive approximation of V by potentials for which the Fermi Golden Rule holds. On the other hand, the results of Subsection 5.2 are more general, but they are more abstract and less constructive.

5.1. Assume that v 0 ∈ C ∞ (R) satisfies the estimates (5.1) |v (j) 0 (x)| = O j x -m 0 -j , x ∈ R, j ∈ Z + , m 0 > 1.
Then condition O ν is valid for every ν ∈ Z + . Moreover, in this case the eigenfunction ψ (see (2.4)) is in the Schwartz class S(R), while the Jost solutions y j (

•; k), j = 1, 2, belong to C ∞ (R) ∩ L ∞ (R).
Suppose that (2.3) holds true, and the discrete spectrum of the operator H consists of a unique eigenvalue λ. Fix m ∈ Z, and q ∈ Z + such that q > m -. Then it is easy to check that we have Im F q,m (2bq + λ) = Evidently, V ⊥ ∈ L 2 Re (R + ; ̺d̺). Fix ε > 0. Applying Lemma 5.1, we find Ṽ⊥ ∈ L 2 Re (R + ; ̺d̺) such that (5.9)

∞ 0 ϕ q-1,m (̺)ϕ q,m (̺) Ṽ⊥ (̺)̺d̺ = 0
for every m ∈ Z, q > m -, and

(5.10)

V ⊥ -Ṽ⊥ L 2 (R + ;̺d̺) < ε. Set (5.11) Ṽ (̺, x 3 ) := Ṽ⊥ (̺)ω(x 3 ) ω 2 L 2 (R) + V (̺, x 3 ) - V ⊥ (̺)ω(x 3 ) ω 2 L 2 (R) , ̺ ∈ R + , x 3 ∈ R.
We have

I q,m,λ ( Ṽ ) = ∞ 0 ϕ q-1,m (̺)ϕ q,m (̺) Ṽ⊥ (̺)̺d̺ = 0
for every m ∈ Z, q > m -. On the other hand, (5.10) and (5.11) imply

V -Ṽ 2 Dν ≤ ε 2 ν j=0 R x 2j ω (j) (x) 2 dx ω 2 L 2 (R)
.

Fix again ν ∈ Z + . We will write V ∈ E ν if V : R + → R is continuous and tends to zero at infinity, and the functions x j 3 ∂ j V (̺,x 3 ) ∂x j

3

, j = 1, . . . , ν, are bounded. If O ν holds, and V ∈ E ν , then C ν holds true. For V ∈ E ν define the norm

V Eν := ν j=0 sup (̺,x 3 )∈R + ×R x j 3 ∂ j V (̺, x 3 ) ∂x j 3 .
Arguing as in the proof of Theorem 5.1, from Lemma 5.2 we obtain the following Theorem 5.2. Assume that v 0 satisfies the hypotheses of Theorem 5.1. Fix ν ∈ Z + . Then the set of perturbations V : R + × R → R for which the Fermi Golden Rule F q,m,λ is valid for each m ∈ Z and q > m -, is dense in E ν .

5.2.

For H a subspace of L r (R), let us introduce the space (5.12)

H † := {ω ∈ S(R) | ∀χ ∈ H, R ω(x)χ(x)dx = 0}. Clearly, if C ∞ b (R) ⊂ H then H † = {0}. This property holds yet if H ∞ (S θ ) ⊂ H where H ∞ (S θ
) is the set of smooth bounded functions on R admitting analytic extension on S θ . It is enough to note that if ω(x 0 ) = 0 then for C sufficiently large, the function

χ(x 3 ) := e -C(x 3 -x 0 ) 2 is in H ∞ (S θ ) and satisfies R ω(x)χ(x)dx = 0. Theorem 5.3. Assume that v 0 satisfies the hypotheses of Theorem 5.1. Let p ≥ 1, r ≥ 1, δ ∈ R.
Let H be a Banach space contained in L r (R) such that the injection H ֒→ L r (R) is continuous, and H † = {0}.

Then the set of real perturbations V : R + × R → R, for which the Fermi Golden Rule F q,m,λ is valid for each m ∈ Z and q > m -, is dense in L p (R + , ̺ δ ̺d̺; H).

Proof. As in the case of Theorems 5.1 -5.2, we will prove that the set of perturbations V for which the integral (5.8) does not vanish for each m ∈ Z and q > m -, is dense in

L p (R + , ̺ δ ̺d̺; H). Let M q,m,λ := {V ∈ L p (R + , ̺ δ ̺d̺; H); I q,m,λ (V ) = 0}.
Since for 1/p ′ + 1/p = 1 and 1/r ′ + 1/r = 1 we have

|I q,m,λ (V )| ≤ ϕ q-1,m ϕ q,m L p ′ (R + , ̺ -δ ̺d̺;R) ω L r ′ (R) V L p (R + , ̺ δ ̺d̺;L r (R)) ,
the continuity of the injection H ֒→ L r (R) implies that M q,m,λ is an open subset of the Banach space L p (R + , ̺ δ ̺d̺; H). Then according to the Baire lemma, we have only to check that each M q,m,λ is dense in L p (R + , ̺ δ ̺d̺; H). Let V ∈ L p (R + , ̺ δ ̺d̺; H) \ M q,m,λ . Since 0 = ω = ω ∈ S(R), the assumptions on H imply the existence of a Φ ∈ H such that R ω(x 3 )Φ(x 3 )dx 3 = 0.

Moreover, ̺ → ϕ q-1,m (̺)ϕ q,m (̺)̺ is a product of polynomial and exponential functions.

Then there exists ̺ 0 ∈ R + such that ϕ q-1,m (̺ 0 )ϕ q,m (̺ 0 )̺ 0 = 0, and for χ 0 supported near ̺ 0 , we have ∞ 0 ϕ q-1,m (̺)ϕ q,m (̺)χ 0 (̺)̺d̺ = 0.

Consequently, V (̺, x 3 ) + 1 n χ 0 (̺)Φ(x 3 ) n∈N is a sequence of functions in M q,m,λ tending to V in L p (R + , ̺ δ ̺d̺; H).

6.

Singularities of the spectral shift function 6.1. Suppose that v 0 satisfies (2.5). Assume moreover that the perturbation V : R 3 → R satisfies (2.11) with m ⊥ > 2 and m 3 = m 0 > 1. Then the multiplier by V is a relatively trace-class perturbation of H. Hence, the spectral shift function (SSF) ξ(•; H + V, H) satisfying the Lifshits-Krein trace formula

Tr(f (H + V ) -f (H)) = R f ′ (E)ξ(E; H + V, H)dE, f ∈ C ∞ 0 (R),
and normalized by the condition ξ(E; H + V, H) = 0 for E < inf σ(H + V ), is welldefined as an element of L 1 (R; E -2 dE) (see [START_REF] Lifshits | On a problem in perturbation theory[END_REF], [START_REF] Krein | On the trace formula in perturbation theory[END_REF]). If E < inf σ(H), then the spectrum of H + V below E could be at most discrete, and for almost every E < inf σ(H) we have

ξ(E; H + V, H) = -rank P (-∞,E) (H + V ).
On the other hand, for almost every E ∈ σ ac (H) = [0, ∞), the SSF ξ(E; H + V, H) is related to the scattering determinant det S(E; H + V, H) for the pair (H + V, H) by the Birman-Krein formula det S(E; H + V, H) = e -2πiξ(E;H+V,H)

(see [START_REF] Birman | On the theory of wave operators and scattering operators[END_REF]).

Set

Z := {E ∈ R|E = 2bq + µ, q ∈ Z + , µ ∈ σ disc (H ) or µ = 0}.
Arguing as in the proof of [9, Proposition 2.5], we can easily check the validity of the following Proposition 6.1. Let v 0 and V satisfy (2.5), and (2.11) with m ⊥ > 2 and m 0 = m 3 > 1.

Then the SSF ξ(•; H + V, H) is bounded on every compact subset of R \ Z, and is continuous on R \ (Z ∪ σ p (H + V )), where σ p (H + V ) denotes the set of the eigenvalues of the operator

H + V .
In what follows we will assume in addition that

(6.1) 0 ≤ V (x), x ∈ R 3 ,
and will consider the operators H ± V which are sign-definite perturbations of the operator H. The goal of this section is to investigate the asymptotic behaviour of the SSF ξ(•; H ± V, H) near the energies which are eigenvalues of H of infinite multiplicity. More precisely, if (2.3) holds true, and λ ∈ σ disc (H ), we will study the asymptotics as η → 0 of ξ(2bq + λ + η; H ± V, H), q ∈ Z + , being fixed. Let T be a compact self-adjoint operator. For s > 0 denote n ± (s; T ) := rank P (s,∞) (±T ), n * (s; T ) := n + (s; T ) + n -(s; T ).

Put

U(X ⊥ ) := R V (X ⊥ , x 3 )ψ(x 3 ) 2 dx 3 , X ⊥ ∈ R 2 ,
the eigenfunction ψ being defined in (2.4).

Theorem 6.1. Let v 0 and V satisfy (2.5), (2.11) with m ⊥ > 2 and m 0 = m 3 > 1, and (6.1). Assume that (2.3) holds true, and λ ∈ σ disc (H ). Fix q ∈ Z + . Then for each ε ∈ (0, 1) we have (6.2)

n + ((1 + ε)η; p q Up q ) + O(1) ≤ ±ξ(2bq + λ ± η; H ± V, H) ≤ n + ((1 -ε)η; p q Up q ) + O(1), (6.3) ξ(2bq + λ ∓ η; H ± V, H) = O(1), as η ↓ 0.
Applying the well known results on the spectral asymptotics for compact Berezin-Toeplitz operators p q Up q (see [START_REF] Raikov | Eigenvalue asymptotics for the Schrödinger operator with homogeneous magnetic potential and decreasing electric potential. I. Behaviour near the essential spectrum tips[END_REF], [START_REF] Raikov | Quasi-classical versus non-classical spectral asymptotics for magnetic Schrödinger operators with decreasing electric potentials[END_REF]), we obtain the following Corollary 6.1. Assume the hypotheses of Theorem 6.1.

(i) Suppose that U ∈ C 1 (R 2 ), and

U(X ⊥ ) = u 0 (X ⊥ /|X ⊥ |)|X ⊥ | -α (1 + o(1)), |X ⊥ | → ∞, |∇U(X ⊥ )| ≤ C 1 X ⊥ -α-1 , X ⊥ ∈ R 2 ,
where α > 2, and u 0 is a continuous function on S 1 which is non-negative and does not vanish identically. Then we have

ξ(2bq + λ ± η; H ± V, H) = ± b 2π X ⊥ ∈ R 2 |U(X ⊥ ) > η (1 + o(1)) = ±η -2/α b 4π S 1 u 0 (s) 2/α ds (1 + o(1)), η ↓ 0, where |.| denotes the Lebesgue measure. (ii) Let U ∈ L ∞ (R 2 ). Assume that ln U(X ⊥ ) = -µ|X ⊥ | 2β (1 + o(1)), |X ⊥ | → ∞,
for some β ∈ (0, ∞), µ ∈ (0, ∞). Then we have

ξ(2bq + λ ± η; H ± V, H) = ±ϕ β (η) (1 + o(1)), η ↓ 0, β ∈ (0, ∞),
where

ϕ β (η) :=      b 2µ 1/β | ln η| 1/β if 0 < β < 1, 1 ln (1+2µ/b) | ln η| if β = 1, β β-1 (ln | ln η|) -1 | ln η| if 1 < β < ∞, η ∈ (0, e -1 ).
(iii) Let U ∈ L ∞ (R 2 ). Assume that the support of U is compact, and that there exists a constant C > 0 such that U ≥ C on an open non-empty subset of R 2 . Then we have

ξ(2bq + λ ± η; H ± V, H) = ±(ln | ln η|) -1 | ln η|(1 + o(1)), η ↓ 0.
Remarks: (i) The threshold behaviour of the SSF for various magnetic quantum Hamiltonians has been studied in [START_REF] Fernández | On the singularities of the magnetic spectral shift function at the Landau levels[END_REF] (see also [START_REF] Raikov | Spectral shift function for Schrödinger operators in constant magnetic fields[END_REF], [START_REF] Raikov | Spectral shift function for magnetic Schrödinger operators[END_REF]), and recently in [START_REF] Briet | Spectral properties of a magnetic quantum Hamiltonian on a strip[END_REF]. The singularities of the SSF described in Theorem 6.1 and Corollary 6.1 are of somewhat different nature since 2bq + λ is an infinite-multiplicity eigenvalue, and not a threshold in the continuous spectrum of the unperturbed operator.

(ii) By the strict mathematical version of the Breit-Wigner representation for the SSF (see [START_REF] Petkov | Breit-Wigner approximation and the distribution of resonances[END_REF], [START_REF] Petkov | Semi-classical estimates on the scattering determinant[END_REF]), the resonances for various quantum Hamiltonians could be interpreted as the poles of the SSF. In [START_REF] Bony | Resonances and spectral shift function near Landau levels[END_REF] a Breit-Wigner approximation of the SSF near the Landau level was obtained for the 3D Schrödinger operator with constant magnetic field, perturbed by a scalar potential satisfying (2.11) with m ⊥ > 2 and m 3 > 1. Moreover, it was shown in [START_REF] Bony | Resonances and spectral shift function near Landau levels[END_REF] that typically the resonances accumulate at the Landau levels. It is conjectured that the singularities of the SSF ξ(•; H ± V, H) at the points 2bq + λ, q ∈ Z + , are due to accumulation of resonances to these points. One simple motivation for this conjecture is the fact that if V is axisymmetric, then the eigenvalues of the operators p q Up q , q ∈ Z + , appearing in (6.2) are equal exactly to the quantities V Φ q,m , Φ q,m L 2 (R + ×R;̺d̺dx 3 ) , m ≥ -q, occurring in (3.1) and (4.12). We leave for a future work the detailed analysis of the relation between the singularities of the SSF at the points 2bq + λ and the eventual accumulation of resonances at these points. Hopefully, in this future work we will also extend our results of Sections 3 -5 to the case of non-axisymmetric perturbations V . (iii) As mentioned above, if λ ∈ σ disc (H ), then λ is an isolated eigenvalue of H of infinite multiplicity. Set

λ -:= sup{µ ∈ σ(H), µ < λ} if λ > inf σ(H), -∞ if λ = inf σ(H), λ + := inf{µ ∈ σ(H), µ > λ}.
By Pushnitski's representation of the SSF (see [START_REF] Pushnitskiȋ | A representation for the spectral shift function in the case of perturbations of fixed sign[END_REF]), and the Birman-Schwinger principle for discrete eigenvalues in gaps of the essential spectrum, we have

ξ(λ -η; H -V ; H) = -n + (1; V 1/2 (H -λ + η) -1 V 1/2 ) = -rank P (λ -,λ-η) (H -V ) + O(1), η ↓ 0, ξ(λ + η; H + V ; H) = n -(1; V 1/2 (H -λ -η) -1 V 1/2 ) = rank P (λ+η,λ + ) (H + V ) + O(1), η ↓ 0.
Then Theorem 6.1 and Corollary 6.1 imply that the perturbed operator H -V (resp., H + V ) has an infinite sequence of discrete eigenvalues accumulating to λ from the left (resp., from the right).

6.2. This subsection contains some preliminary results needed for the proof of Theorem 6. [START_REF] Agmon | Spectral properties of Schrödinger operators and scattering theory[END_REF].

In what follows we denote by S 1 the trace class, and by S 2 the Hilbert-Schmidt class of compact operators, and by • j the norm in S j , j = 1, 2. Suppose that η ∈ R satisfies (6.4) 0 < |η| < min 2b + λ, 1 2 dist λ, σ(H ) \ {λ} .

Note that inequalities (6.4) combined with (2.3), imply

λ + η ∈ (-2b, 0), λ + η ∈ σ(H ), dist (λ + η, σ(H )) = |η|. Set P j = p j ⊗ I , j ∈ Z + . For z ∈ C + := {ζ ∈ C|Im ζ > 0}, j ∈ Z + , and
W := V 1/2 , put T j (z) := W P j (H -z) -1 W.
Proposition 6.2. Assume the hypotheses of Theorem 6.1. Suppose that (6.4) holds true. Fix q ∈ Z + . Let j ∈ Z + , j ≤ q. Then the operator-norm limit (6.5)

T j (2bq + λ + η) = lim δ↓0 T j (2bq + λ + η + iδ) exists in L(L 2 (R 3
)). Moreover, if j < q, we have T j (2bq + λ + η) ∈ S 1 , and

(6.6) T j (2bq + λ + η) 1 = O(1), η → 0.
Proof. We have (6.7)

T j (z) = M(t ⊥,j ⊗ t (z -2bj))M, z ∈ C + , where M := W (X ⊥ , x 3 ) X ⊥ m ⊥ /2 x 3 m 3 /2 , t ⊥,l := X ⊥ -m ⊥ /2 p l X ⊥ -m ⊥ /2 , l ∈ Z + , t (ζ) := x 3 -m 3 /2 (H -ζ) -1 x 3 -m 3 /2 , ζ ∈ C + .
Since the operators M and t ⊥ are bounded, in order to prove that the limit (6.5) exists in L(L 2 (R 3 )), it suffices to show that the operator-norm limit (6.8) lim δ↓0 t (2b(qj) + λ + η + iδ) exists in L(L 2 (R)). If j < q, the limit in (6.8) exists due to the existence of the limit in (2.6). If j = q, the limit in (6.8) exists just because λ + η ∈ σ(H ). Further, set

t ,0 (ζ) := x 3 -m 3 /2 (H 0, -ζ) -1 x 3 -m 3 /2 , ζ ∈ C + \ {0}.
For E = 2b(qj) + λ + η, from the resolvent equation we deduce (6.9)

t (E) = t ,0 (E)(I -Mt (E))
where M := v 0 (x 3 ) x 3 m 3 is a bounded multiplier. By [9, Section 4.1], the operator t ,0 (E) with E ∈ R \ {0} is trace-class, and we have with c independent of E. Assume j < q. Then (6.9), (6.10), and (2.7) imply t (2b(qj) + λ + η) ∈ S 1 , and (6.11) t (2b(qj) + λ + η) 1 = O(1), η → 0.

Finally, for any l ∈ Z + we have t ⊥,l ∈ S 1 , and (6.12)

t ⊥,l 1 = b 2π R 2 X ⊥ -m ⊥ dX ⊥
(see e.g. [9, Subsection 4.1]). Bearing in mind the structure of the operator T j (see (6.7)) and the boundedness of the operator M, we find that T j (2b(qj) + λ + η) ∈ S 1 , and due to (6.11) and (6.12), estimate (6.6) holds true. Now set P + q := ∞ j=q+1 p j , q ∈ Z + , the convergence of the series being understood in the strong sense. For z ∈ C + set T + q (z) := W (Hz) -1 P + q W. Proposition 6.3. Assume that v 0 , V , and λ, satisfy the hypotheses of Theorem 6.1, and η ∈ R satisfies (6.4). Fix q ∈ Z + . Then the operator-norm limit (6.13) T + q (2bq + λ + η) = lim δ↓0 T + q (2bq + λ + η + iδ) exists in L(L 2 (R 3 )). Moreover, T + q (2bq + λ + η) ∈ S 2 , and (6.14)

T + q (2bq + λ + η) 2 = O(1), η → 0. Proof. Due to (2.3), the operator-valued function

C + ∋ z → (H -z) -1 P + q → L(L 2 (R 3
)) admits an analytic continuation in {ζ ∈ C | Re ζ < 2bq}. Since λ + η < 0, and W is bounded, we immediately find that the limit in (6.13) exists. Evidently, the operatorvalued function C + ∋ z → (H 0z) -1 P + q → L(L 2 (R 3 )) also admits an analytic continuation in {ζ ∈ C | Re ζ < 2bq}, and for E = 2bq + λ + η we have (6.15)

T + q (E) = W (H 0 -E) -1 P + q (Wv 0 (H -E) -1 P + q W ). Arguing as in the proof of [START_REF] Fernández | On the singularities of the magnetic spectral shift function at the Landau levels[END_REF]Proposition 4.2], we obtain W (H 0 -2bqλη) -1 P + q ∈ S 2 , and (6. [START_REF] Fernández | On the singularities of the magnetic spectral shift function at the Landau levels[END_REF])

W (H 0 -2bqλη) -1 P + q 2 = O(1), η → 0. Since λ < 0, we have (6.17)

Wv 0 (H -2bqλη) -1 P + q W = O(1), η → 0. Putting together (6.15) and (6.16)-(6.17), we obtain (6.14).

6.3.

In this subsection we prove Theorem 6.1. Suppose that η ∈ R satisfies (6.4). Fix q ∈ Z + . Set T (2bq + λ + η) := T - q (2bq + λ + η) + T q (2bq + λ + η) + T + q (2bq + λ + η), where T - q (2bq + λ + η) = j<q T j (2bq + λ + η).

Note that the operators T q (2bq + λ + η) and T + q (2bq + λ + η) are self-adjoint. By Pushnitski's representation of the SSF for sign-definite perturbations (see [START_REF] Pushnitskiȋ | A representation for the spectral shift function in the case of perturbations of fixed sign[END_REF]), we have (6.18) ξ(2bq + λ + η; H ± V, H) = ± 1 π R n ∓ (1, Re T (2bq + λ + η) + s Im T (2bq + λ + η)) ds 1 + s 2 . By (6.18) and the well-known Weyl inequalities, for each ε ∈ (0, 1) we have n ∓ (1 + ε; T q (2bq + λ + η)) -R ε (η) ≤ ±ξ(2bq + λ + η; H ± V, H) ≤ (6. [START_REF] Herbst | Exponential decay in the Stark effect[END_REF] n ∓ (1ε; T q (2bq + λ + η)) + R ε (η) where R ε (η) := n * (ε/3; Re T - q (2bq + λ + η)) + n * (ε/3; T + q (2bq + λ + η)) + 3 ε T - q (2bq + λ + η) 1 ≤ (6.20) 6 ε j<q T j (2bq + λ + η) 1 + 9 ε 2 T + q (2bq + λ + η) 2 2 = O(1), η → 0, due to Propositions 6.1 -6.2. Next, set τ q = W (p q ⊗ p )W, Tq (λ + η) := W (p q ⊗ (Hλη) -1 (Ip ))W, provided that η ∈ R satisfies (6.4). Evidently, T q (2bq + λ + η) = -η -1 τ q + Tq (λ + η).

Applying again the Weyl inequalities, we get n ± (s(1 + ε)|η|; (sign η)τ q )n * (sε; Tq (λ + η)) ≤ n ∓ (s; T q (2bq + λ + η)) ≤ (6.21) n ± (s(1ε)|η|; (sign η)τ q ) + n * (sε; Tq (λ + η))

for each s > 0, ε ∈ (0, 1), and η satisfying (6.4). Note that since p q Up q ≥ 0 we have (6.22) n ± (s|η|; (sign η) τ q ) = n + (s|η|; p q Up q ) if ± η > 0, 0 if ± η < 0, for every s > 0. Further, (6.23) Tq (λ + η) = M(t ⊥,q ⊗ t (λ + η))M where t (λ + η) := x 3 -m 3 /2 (Hλη) -1 (Ip ) x 3 -m 3 /2 .

Obviously, (6.24) t (λ + η) ≤ (Hλη) -1 (Ip ) = O(1), η → 0.

On the other hand, similarly to (6.9) we have (6.25) t (λ + η) = t ,0 (λ + η)(I -M t (λ + η))x 3 -m 3 /2 (H 0,λη) -1 p x 3 -m 3 /2 .

Since p x 3 -m 3 /2 is a rank-one operator, we have Putting together (6.25), (6.10), (6.24), and (6.26), we get (6.27) t (λ + η) 1 = O(1), η → 0, which combined with (6.23) and (6.12) yields (6.28) n ± (s; Tq (λ + η)) ≤ s -1 Tq (λ + η) = O(1), η → 0. Now (6.2) -(6.3) follow from estimates (6.19) -(6.22), and (6.28).

x 3 -m 3 / 2 (p x 3 -m 3 / 2 1 ≤ x 3 -m 3 / 2 (p x 3 -

 332323323 H 0,λη) -1 H 0,λη) -1 3 ψ(x) 2 dx |λ + η| -1 = O(1), η → 0.

Acknowledgements. M. A. Astaburuaga and C. Fernández were partially supported by the Chilean Laboratorio de Análisis Estocástico PBCT -ACT 13. Philippe Briet and Georgi Raikov were partially supported by the CNRS-Conicyt Grant "Resonances and embedded eigenvalues for quantum and classical systems in exterior magnetic fields". V. Bruneau was partially supported by the French ANR Grant no. JC0546063. V. Bruneau and G. Raikov were partially supported by the Chilean Science Foundation Fondecyt under Grants 1050716 and 7060245.

ϕ j,m (̺)ϕ q,m (̺)ψ(x 3 )Ψ l (x 3 ; 2b(qj) + λ)V (̺, x 3 )dx 3 ̺d̺

2

.

In what follows we denote by L 2 Re (R + ; ̺d̺) the set of real functions W ∈ L 2 (R + ; ̺d̺). Lemma 5.1. The set of functions W ∈ L 2 Re (R + ; ̺d̺) for which

Proof. Since the Laguerre polynomials L (0) q , q ∈ Z + , (see (2.2)) form an orthogonal basis in L 2 (R + ; e -s ds), the set of polynomials is dense in L 2 (R + ; e -s ds). Pick W ∈ L 2 Re (R + ; ̺d̺). Set w(s) := W ( 2s/b)e s/2 , s > 0. Evidently, w = w ∈ L 2 (R + ; e -s ds). Pick ε > 0 and find a non-zero polynomial P with real coefficients such that ∞ 0 e -s (P(s)w(s)) 2 ds < bε 2 4 .

Note that the coefficients of P could be chosen real since the coefficients of the Laguerre polynomials are real. Changing the variable s = b̺ 2 /2, we get (5.4)

, where the real polynomial P is fixed and satisfies (5.4). We will show that the set (5.5)

is dense in (0, ∞). Actually, for fixed m ∈ Z and q > m -, we have

where Π q,m is a real polynomial of degree 2q + m + 1 + deg P, and γ(α

Note that γ : (0, ∞) → (0, 1) is a bijection. Denote by N q,m the set of the zeros of Π q,m lying on the interval (0, 1). Set

Evidently, the sets N and γ -1 (N ) are countable, and A = (0, ∞) \ γ -1 (N ). Therefore, A is dense in (0, ∞). Now, pick α 0 ∈ A so close to 1/2 that (5.6)

Assembling (5.4) and (5.6), we obtain

Denote by K(R) the class of real-valued continuous functions u : [0, ∞) → R such that lim s→∞ u(s) = 0. Set u K(R) := max s∈[0,∞) |u(s)|.

Lemma 5.2. The set of functions W ∈ K(R) for which (5.3) holds true for every m ∈ Z, q > m -, is dense in K(R).

Proof. By the Stone-Weierstrass theorem for locally compact spaces, we find that the set of functions e -αs P(s), s > 0 where α ∈ (0, ∞), and P is a polynomial, is dense in K(R).

Let W ∈ K(R); then we have u ∈ K(R) where u(s) := W ( 2s/b), s > 0. Pick ε > 0 and find α ∈ (0, ∞) and a polynomial P such that W -W α K(R) < ε/2 where, as in the proof of Lemma 5.1, W α (̺) = e -αb̺ 2 /2 P(b̺ 2 /2). Next pick α 0 ∈ A (see (5.5)) such that

Note that if O ν holds, and V ∈ D ν , then C ν is valid.

Theorem 5.1. Assume that:

• v 0 ∈ C ∞ (R) satisfies (5.1);

• inequality (2.3) holds true;

• we have σ disc (H ) = {λ}.

Fix ν ∈ Z + . Then the set of real perturbations V : R + × R → R for which the Fermi Golden Rule F q,m,λ is valid for each m ∈ Z and q > m -, is dense in D ν .

Proof. We will prove that the set of perturbations V for which the integral (5.8)

̺)ϕ q,m (̺)ψ(x 3 )Ψ 1 (x 3 ; 2b + λ)V (̺, x 3 )dx 3 ̺d̺ does not vanish for each m ∈ Z and q > m -, is dense in D ν . By (5.2) this will imply the claim of the theorem. Set ω(x) := ψ(x)Re Ψ 1 (x; 2b + λ), x ∈ R.

Note that 0 = ω = ω ∈ S(R). Set V ⊥ (̺) = R ω(x 3 )V (̺, x 3 )dx 3 , ̺ ∈ R + .


Natalia Díaz-Arce

Peter Grewe

Iñigo Krug

Iraide Artetxe

Jon Ruiz

Natacha Nikolic

Anais Medieu

Marianne Pernak

Matt Lansdell

Jorden Aulich

Naomi Clear

Craig Proctor

Toni Ruchimat

Zulkarnaen Fahmi

Fayakun Satria

Pratiwi Lestari

Muhammad Taufik

Asep Priatna

Achmad Zamroni

Jessica Farley

Campbell R Davies

Francis Marsac

Igaratza Fraile

Hilario Murua

Naiara Rodríguez-Ezpeleta

Evidence of connectivity of bigeye tuna (Thunnus obesus) throughout the Indian Ocean inferred from genome-wide genetic markers

In order to resolve the population connectivity of bigeye tuna (Thunnus obesus) within the Indian Ocean, we analyzed thousands of genome-wide markers of individuals from a broad geographic area of the Indian Ocean as well as from locations in the Pacific and Atlantic Oceans. Our results support a single panmictic population of bigeye tuna within the Indian Ocean isolated from the Atlantic and Pacific Oceans.

Introduction

Previous studies on the population structure of bigeye tuna (Thunnus obesus) support inter-oceanic genetic separation [START_REF] Bremer | Genetic evidence for interoceanic subdivision of bigeye tuna (Thunnus obesus) populations[END_REF][START_REF] Chow | Genetic divergence between Atlantic and Indo-Pacific stocks of bigeye tuna (Thunnus obesus) and admixture around South Africa[END_REF][START_REF] Durand | Nuclear and mitochondrial DNA markers indicate unidirectional gene flow of Indo-Pacific to Atlantic bigeye tuna (Thunnus obesus) populations, and their admixture off southern Africa[END_REF][START_REF] Martínez | Genetic diversity and historical demography of Atlantic bigeye tuna (Thunnus obesus)[END_REF]). However, the few studies that applied genetic methods to understanding population structure of this species within the Indian Ocean did not observe signs of heterogeneity, supporting the existence of a single panmictic IOTC-2020-WPTT22(AS)-16 population [START_REF] Appleyard | Genetic stock structure of bigeye tuna in the Indian Ocean using mitochondrial DNA and microsatellites[END_REF][START_REF] Chiang | Population structure of bigeye tuna (Thunnus obesus) in the Indian Ocean inferred from mitochondrial DNA[END_REF]. In order to resolve population structure of bigeye tuna within the Indian Ocean, we have analyzed thousands of Single Nucleotide Polymorphisms (SNP) markers from individuals collected throughout the distribution of the species.

Materials and methods

Sampling, DNA extraction and RAD-seq library preparation

Bigeye tuna muscle tissue samples were obtained from a broad geographic area of the Indian Ocean, together with additional samples from the Pacific and Atlantic Oceans were obtained, by scientific observers on-board purse seiner vessels or by sampling fish during landings at port (Figure 1). In total, 496 fish were sampled from 10 areas (Maldives, Central Indian Ocean, East Central Indian Ocean, North Central Indian Ocean, North East Indian Ocean, South East Indian Ocean, South West Indian Ocean, West Central Indian Ocean, West Tasman Sea and East Atlantic Ocean). Samples were from fish classified by Straight Fork Length (SFT) as young of the year (YoY) (<45 cm), juveniles (45-120 cm) and adults (>120 cm) according to according to previously described agelength relationship [START_REF] Eveson | Estimating growth of tropical tunas in the Indian Ocean using tag-recapture data and otolith-based age estimates[END_REF][START_REF] Sardenne | Determining the age of tropical tunas in the Indian Ocean from otolith microstructures[END_REF] and maturity thresholds [START_REF] Zudaire | Sex-ratio, size at maturity, spawning period and fecundity of bigeye tuna (Thunnus obesus) in the western Indian Ocean[END_REF]. DNA was extracted from about 15 mg of tissue on an Eppedorf EP motion 5057 liquid robotic handler using a modification of the QIAamp® 96 DNA QIAcube HT Kit (QIAGEN, Hilden, Germany). This extraction includes a lysis step in the presence of Proteinase K followed by bind-wash-elute QIAGEN technology. DNA sample libraries were created in digestion/ligation reactions using two restriction enzymes, PstI and SphI. The PstI site was compatible with a forward adapter that included an Illumina flow cell attachment sequence and a sequencing primer sequence incorporating a "staggered", varying length barcode region. SphI-generated a compatible overhang sequence that was ligated to a reverse adapter containing a flow cell attachment region and reverse priming sequence. Only "mixed fragments" (PstI-SphI) were effectively amplified by PCR. PCR conditions consisted of an initial denaturation at 94°C for 1 min followed by 30 cycles of 94°C for 20 sec, 58°C for 30 sec and 72°C for 45 sec, with a final extension step at 72°C for 7 min. After PCR, equimolar amounts of amplification products from each sample of the 96-well microtiter plate were bulked and applied to cBot (Illumina) bridge PCR, followed by sequencing on an Illumina Hiseq2000. The sequencing (single read) was run for 77 cycles.

RAD-tag assembly and SNP calling

Generate Dart-seq reads were analyzed using Stacks version 2.4 [START_REF] Catchen | Stacks: an analysis tool set for population genomics[END_REF].

Using process_radtags, reads were truncated to 69bp so that all reads had the same length after barcode removal and reads with any uncalled base, with total low-quality scores or with quality score below 20 within 10bp size sliding windows were removed.

The module ustacks was then used to assemble orthologous tags (stacks) per individual, with a minimum coverage depth required to create a stack (parameter -m) of 3, and a maximum nucleotide mismatches allowed between stacks (parameter -M) of 2. Matches to the catalog for each sample were searched using sstacks and transposed using tsv2bam and the module gstacks was used to identifying and genotyping SNPs. Only samples with more than 30,000 and less than 65,000 reads were selected for further analyses. The module populations was used to export from the catalog, the SNPs presented in RAD loci found in at least 75% of the individuals. Using PLINK version 1.07 [START_REF] Purcell | PLINK: A Tool Set for Whole-Genome Association and Population-Based Linkage Analyses[END_REF], SNPs with more than 5% missing data and a minimum allele frequency (MAF) smaller than 0.05 or failing the Hardy-Weinberg equilibrium test (p < 0.05) in at least two location (excluding North Central Indian (NCI), Central Indian Ocean (CIO) and Maldives (CIM) locations which contained less than 10 individuals), as well as samples with more than 10% missing data were excluded from downstream analyses.

Finally, only the first SNP per loci were kept and resulting genotype table was exported into Structure and Genepop formats. Related individuals were identified using GCTA [START_REF] Yang | GCTA: A Tool for Genome-wide Complex Trait Analysis[END_REF]).

Genetic diversity and population structure analyses

Related individuals were identified using GCTA [START_REF] Yang | GCTA: A Tool for Genome-wide Complex Trait Analysis[END_REF]. Principal Component Analysis (PCA) were performed using the adegenet R package [START_REF] Jombart | adegenet 1.3-1: new tools for the analysis of genome-wide SNP data[END_REF], and ADMIXTURE [START_REF] Alexander | Fast model-based estimation of ancestry in unrelated individuals[END_REF] was run assuming from 1 to 6 ancestral populations (K) setting default parameters. The value of K with lowest associated error value was identified using ADMIXTURE's cross-validation procedure.

Results

Principal component analyses (Figure 2) based on the filtered dataset consisting of 472 individuals and 6,093 SNPs show strong differentiation between samples from the three oceans, but do not suggest any intra-oceanic structure within the Indian Ocean, even when including only samples from the Indian Ocean. ADMIXTURE analyses show that the number of assumed ancestral populations with the lowest associated error (K) is 2 (Figure 3) and show three clusters corresponding to the three oceans.

Main conclusions

Our findings support the hypothesis that bigeye from Atlantic, Indian and Pacific oceans form three genetically distinct populations. Within the Indian ocean, no genetic differentiation can be observed suggesting that the bigeye from the Indian Ocean forms a single panmictic population.

Fig. 1 .

 1 Fig. 1. Samples collected for this study. Each location is represented by one color (CIM -Maldives, CIO -Central Indian Ocean, ECI -East Central Indian Ocean, NCI -North Central Indian Ocean, NEI -North East Indian Ocean, SEI -South East Indian Ocean, SWI -South West Indian Ocean, WCI -West Central Indian Ocean, WTS -West Tasman Sea and EAO -East Atlantic Ocean) and shapes indicate whether samples are young of the year (YoY), juveniles or adults. Size of shapes are proportional to the number of samples collected per area/age.

Fig. 2 .

 2 Fig. 2. Principal Component Analysis (PCA) performed using the final dataset containing all samples (left) or only the Indian Ocean samples (right). Different colors in the PCA represent samples from the different locations at which samples were captured.

Fig. 3 .

 3 Fig. 3. Individual ancestry proportions estimated by ADMIXTURE when assuming from two to four ancestral populations (K) when including all samples (top) or only Indian Ocean samples (bottom).

Acknowledgements

This work is part of the PSTBS project supported by funding from CSIRO Oceans and Atmosphere, AZTI Basque Research and Technology Alliance, Institut de recherche pour le développement (IRD), and Research Institute for Tuna Fisheries (RITF) and financial assistance of the European Union (GCP/INT/233/EC -Population structure of IOTC species in the Indian Ocean). The views expressed herein can in no way be taken to reflect the official opinion of the European Union. We would like to thank the many vessel owners, skippers, observers, crews and processors who provided fish for sampling and access to facilities.