SEMICLASSICAL RESONANCES VIA MEROMORPHY OF THE RESOLVENT AND THE S-MATRIX

The purpose of this paper is to describe the basic problems of resonances via meromorphic continuation of the resolvent and the scattering matrix. An example from mathematical physics is given by investigating the poles of the resolvent of semiclassical Schrödinger operators and Born-Oppenheimer Hamiltonians. Mathematical techniques, dilation-analyticity and Feshbach reduction are used here for the characterization of resonances of these Hamiltonians.


Introduction
The spectrum in the complex plane of Schrödinger operators P (h) = −h 2 ∆ + V, is often the union of the line Imz = 0 and at most finitely-many points of the form iλ j (h) on the positive imaginary axis λ j (h) > 0. These points correspond to the negative eigenvalues of P (h) so that z(h) = −λ 2 j (h) belongs to the discrete spectrum of P (h). The resolvent P (h) − λ 2 −1 is an operator-valued function defined for Imλ > 0 and λ = λ j (h). We would like to find the largest region in the complex λ-plane on which the resolvent can be defined. For several types of potential V , the spectrum of −h 2 ∆ + V is continuous and equals [0, +∞[, and hence contains no (further) information about V. In this setting resonances replace the discrete data of eigenvalues. Precisely, the poles of the meromorphic continuation of the resolvent are called resonances or scattering poles. They constitute a natural remplacement of discrete spectral data for problems on non-compact domains. The multiplicity of a pole λ 0 is given in terms of multiplicity of the corresponding resonance z 0 = λ 2 0 , multiplicity of λ 0 = dim Imπ λ 2 0 L 2 comp (R n ) where: The resonances are shown to be the same as the poles of the meromorphically continued scattering matrix.
Meromorphic extensions of resolvents have been studied in many frameworks and their finite rank poles or resonances, serve in a sense as discrete data similar in character to eigenvalues of a compact manifold.
However, if the manifold has constant negative sectional curvature away from a compact, Guillopé and Zworski [1] showed the meromorphic continuation of the resolvent to C with finite rank poles. For n odd they are defined as the poles of the meromorphic continuation of P (h) − λ 2 −1 : L 2 comp (R n ) −→ L 2 loc (R n ) from {Imλ > 0} to C or to the Riemann surface (logarithmic covering of C) if n is even. The main advantage of odd dimensions greater than one is the strong Huyghens principle for the wave equation. Effectively, one consequence of the strong Huyghens principle is the analytic continuation of −h 2 ∆ − λ 2 −1 from {Imλ > 0} to C.
Under suitable assumptions on V, the operator P (h) extends as a selfadjoint family of operators on the algebra of bounded operators on L 2 (R n ). As operator on L 2 (R n ), R V (λ) has no analytic extension across its spectrum. But, if we replace L 2 (R n ) by a smaller dense subspace, like C ∞ 0 (R n ), then R V (λ) might have some continuation across [0, +∞[ to some Riemann surface above C [0, +∞[. If the continuation turns out to be meromorphic, we then obtain the resonance of P (h) which are exactly the poles of this continuation.
When V = 0, i.e. P (h) = −h 2 ∆, is the free Hamiltonian, the resonances can be accessible using Fourier analysis. If V = 0, many effective approaches combine the known extension of the free resolvent to properties of V.
The mathematical study of resonances initiated for Schrödinger operators on R n . Later, it was extended to more geometric situations, such as the Laplacian on hyperbolic and asymptotically hyperbolic manifolds, symmetric or locally symmetric spaces, and Damek-Ricci spaces, see e.g. [1] and [2]. In a typical situation, one works on a complete Riemannian manifold X, for which the positive Laplacian −∆ is an essentially self-adjoint operator on the Hilbert space L 2 (X) of square integrable functions on X.
The basic problems of resonances are described here for Schrödinger operators and Born-Oppenheimer Hamiltonians with regular and singular potentials. We first show an holomorphic extension result via Fredholm operator theory in Hilbert spaces. In section 2, we review basic situations for meromorphic continuation of the resolvent. We study in section 3, the meromorphy of the scattering matrix, it follows that the poles of the meromorphic continuation of the S-matrix are exactly the poles of the continuation of the resolvent and conversely. Some interesting characterizations of the resonances of semiclassical Schrödinger operators and Hamiltonians in the Born-Oppenheimer approximation are obtained in section 4, by dilation-analyticity and Feshbach reduction.
We start with the following definition. if for each λ 0 ∈ Ω, there exist a neighbourhood U λ0 of λ 0 , an integer p > 0 and some (A i ) 1≤i≤p ⊂ B(H) such that for all λ ∈ U λ0 {λ 0 } , we have the finite Laurent expansion: where B(λ) is an holomorphic function on U λ0 with values in the algebra B(H) of bounded linear operators on H.
It is easy to see that A(λ) is holomorphic in U S where S is a discrete set of U whose elements are the poles of A(λ). p is the order of the pole, A 1 is the residue of A(λ) at λ 0 .
We essentially have the following result: Proof. For any z ∈ Ω, let n + = dim ker A(z) and n − = dim H ImA(z) and set H + = C n+ and H − = C n− . Let e 1 , ..., e n+ be a basis of H + . So define Next choose y 1 , ..., y n− whose images in H ImA(z) form a basis of H ImA(z) and define We produce a Grushin problem for A(z) as decribed by Belmouhoub and Messirdi in [3]: A(z) is well-posed Grushin problem for z in some sufficiently small neighborhood V (z) of z with inverse In particular, this is true for z = z 0 and for z ∈ V (z). We know that the index of A(z) is constant in V (z). Since the index vanishes at z 0 , then at any z we have n + = n − = n and E −+ (z) is an n × n matrix with holomorphic coefficients. So for any z ∈ Ω the function Since the classical theorems on the usual Fourier transform extend to the semiclassical case, it suffices to consider h = 1. Then the existence of R 0 (λ) follows from using the Fourier transform which provides an explicit diagonalization of −∆ : is of course valid in all dimension but the resolvent operator R 0 (λ) has much nice properties when n is odd. We establish the following important result concerning the meromorphic continuation of the resolvent in odd dimensions.
Imλ > 0, extends analytically to an entire family of operators from L 2 comp (R n ) to L 2 loc (R n ). For any χ ∈ C ∞ 0 (R n ) : Proof. By the functional calculus, we have (see e.g. [4]): is the solution of the wave equation with the initial conditions u(x, 0) = φ 0 (x) and ∂u ∂t (x, 0) = φ 1 (x). In odd dimensions, the strong Huyghens principle (see [5]) implies that: The right hand side is now defined and, as an operator L 2 (R n ) −→ L 2 (R n ), holomorphic for λ ∈ C. In fact, integration by parts shows that the right hand side is bounded with a bound depending on R and α = max (−Imλ, 0) . So, We can consider U (t) as a map from L 2 (R n ) to the Sobolev space H 1 (R n ). Indeed, since sup λ∈R sin tλ λ = |t| , we have: (1) and integrating shows that: We also get a bound for χR 0 (λ)χ as a map from L 2 (R n ) to H 2 (R n ). Recall that the norm on H 2 (R n ) can be taken as u L 2 + ∆u L 2 . So, we have: as operators on functions of compact support), we have: Since [∆, χ] is a first order operator, we obtain: χR 0 (λ)χ is a bounded operator from L 2 (R n ) to L 2 (R n ) and its image consists of functions supported in B(0, R). By Rellich's lemma, the embedding of this space in L 2 (R n ) is compact. Hence: Remark 2.1. Suppose n is odd and R 0 (λ) : Then the analytic continuation of the Schwartz kernel R 0 (λ, x, y) is given by Stone's formula [6]: where dω denotes the standard measure on the unit sphere S n of R n .

Schrödinger operators.
Let's study now the resolvent of the Schrödinger operator P = −∆ + V on exists at points λ ∈ C such that Imλ 0.
Theorem 2.2. (Meromorphic continuation of the resolvent) Suppose that V ∈ L ∞ comp (R n , C) (ie V is a.e. bounded potential of compact support), n ≥ 3 is odd. Then 1) R V (λ) : L 2 (R n ) −→ L 2 (R n ) for Imλ > 0, is a meromorphic family of operators with finitely many poles.
2) R V (λ) extends to a meromorphic family of operators R V (λ) : Proof. We write: where χ ∈ C ∞ 0 (R n ) such that χV = V. Multiply the above equation on the right by χ to get

Meromorphy of the scattering matrix
We have just studied above the meromorphic continuation of the resolvent to C for odd dimensions and to Λ for even dimensions. From this it can be deduced that the scattering matrix has a similar continuation. Indeed, the meromorphic continuation of the cut-off resolvent χR V (λ)χ permits us to mermorphically continue the scattering matrix S(λ) as a bounded operator on L 2 loc (S n ) on C or on Λ depending on the parity of n. In this section, we will define and describe the scattering matrix of P (h) = −h 2 ∆ + V for V ∈ L ∞ comp (R n , R), n ≥ 3, where L ∞ comp (R n , R) is the space of real essentially bounded functions of compact support. P (h) is self-adjoint with domain H 2 (R n ) and generates a one-parameter strongly continuous uni- U 0 (t) is the one-parameter strongly continuous unitary group associated to −h 2 ∆. transformations Ω ± (P (h), −∆) or Ω ± called wave operators: 2) For any f, g ∈ L 2 (R n ) , 3) The operator F ± = Ω ± Ω * ± satisfies: Ω ± U 0 (t) = U V (t)Ω ± and U 0 (t)Ω * ± = Ω * ± U V (t).
Proof. The existence of wave operators comes from an explicit estimate for the free propagation given by U 0 (t) (see e.g. [7]). The relations (2), (3) and (4) follow from the existence of Ω ± and the simple properties of the unitary evolution groups.
The existence of the wave operators Ω ± gives the limits lim It is a bounded operator on L 2 (R n ) since Furthermore, the S-operator commutes with the free time evolution U 0 (t) : This allows for a reduction of the S-operator to a family of operators S(λ) defined on L 2 (S n ) called the S-matrix. Effectively, for λ ∈ R we can define the scattering operator S (λ) : L 2 (S n ) −→ L 2 (S n ) by (see [2]): The resonances do not depend on θ and they are associated with the poles of the meromorphic extension from the upper complex half-plane of the resolvent of P θ (h). In order to prove the existence of such continuation we operate an explicit construction assuming appropriate conditions on the potential.
Let the potential V (x) be smooth real function, extends analytically in |Imθ| < δ 0 , δ 0 > 0, and such that V (−∆ + 1) −1 is compact. We introduce the resonances of P (h) = −h 2 ∆ + V (x) on L 2 (R n ) by using the analytic dilation operator: U θ has a unitary extension on L 2 (R n ). It follows that V (e θ x)(−∆ + 1) −1 is a compact operator-valued analytic function of θ in the strip |Imθ| < δ 0 . Then, is an analytic family of non selfadjoint operators where θ runs in the strip |Imθ| < δ 0 , since for z ∈ C + and ϕ, ψ ∈ L 2 (R n ), Let σ ess = σ σ disc be the essential spectrum where the discrete spectrum σ disc is the set of isolated points of the spectrum such that the corresponding Riesz projectors are finite dimensional. Then, by Weyl Theorem: The definition of resonances is adapted here as follows: (P (h)) and if there exists θ small enough, Imθ > 0, such that ρ ∈ σ disc (P θ (h)). Σ(h) denotes the set of resonances of P (h).

Resonances of Born-Oppenheimer Hamiltonians. The quantum Hamiltonian in the Born-
Oppenheimer approximation is written as: on L 2 (R n x × R p y ) when h tends to 0 + . ∆ x (resp. ∆ y ) is the Laplace operator with respect to x (resp. y), x ∈ R n and y ∈ R p , n = 3m, p = 3q, m, q ≥ 1. Q(x) is the electronic Hamiltonian defined on L 2 (R p y ). The purpose of this section is to show that using a general dilation operator: and the Feshbach reduction scheme, the study of resonances of P (h) is reduced to the discrete spectrum of a matrix of operators F θ (z) defined on L 2 (R n x ) ⊕ L 2 (R n x ) (the so-called effective Hamiltonian) such that: z is a resonance of P (h) ⇐⇒ ∃θ ∈ C, Imθ > 0, z ∈ σ disc F θ (z) .

The dilated Hamiltonian is
Assume that: (H1) V ∈ L ∞ (R n x × R p y , R) and can be analytically extended on the complex strip: Thus, P (h) and Q(x) are selfadjoint on their respective natural domains H 2 (R n x × R p y ) and H 2 (R p y ). In particular, the domain of Q(x) is independent of x. We suppose furthermore: (H2) For every x ∈ R n , the spectrum of Q(x) contains at least two eigenvalues λ 1 (x) and λ 2 (x) where λ 2 (x) is simple and satisfies: In particular, this last assumption implies that the spectral projector π(x) of Q(x) associated to to the wave packet {λ 1 (x), λ 2 (x)} is C 2 -regular with respect to x (see [12]). Furthermore, by the mini-max principle, we deduce that λ 1 (x) and λ 2 (x) are uniformly bounded with respect to x and can be analytically extended on D δ0 .
We also assume that λ 2 (x) has a potential well above the maximum level of λ 1 (x) : By virtue of (H1), Q(e θ x) and P θ (h) can be extended to small enough complex values of θ as analytic families. We then put a Viriel hypothesis to avoid the resonances coming from the effective potential λ 1 (x) near level 0 : Let u j (x, y), j = 1, 2, the two eigenfunctions of Q(x) associated to λ 1 (x) and λ 2 (x) respectively, real and normalized in L 2 (R p y ). We then consider a Grushin problem that will lead to the Feshbach reduction ( [3]). For z, θ ∈ R, let A θ z (h) the matrix operator defined from where u θ j (x, y) = u j (xe θ , y), j = 1, 2, and ., . y denotes the inner product in L 2 (R p y ). We shall study the extension of A θ z (h) to z and θ complexes, for this we have the following results: and by the assumption (H1) : uniformly with respect to x, y and θ complex such that |θ| small enough. Furthermore, for j = 1, 2, and u j extends into a holomorphic function on D δ0 with values in H 2 (R p y ), such that: We now define on L 2 (R n x × R p y ), for θ complex, |θ| small enough, the projector π θ by: π θ u = u, u θ 1 y u θ 1 + u, u θ 2 y u θ 2 and π θ = 1 − π θ . It is therefore simple to show, using previous results, that there is a constant C > 0 such that for θ complex, |θ| small enough, Re π θ (P θ (h) − z) π θ u, π θ u y ≥ C π θ u 2 (4.4) for all u ∈ L 2 (R n x × R p y ) and |z| small enough. In particular, the estimate (4.4) shows the existence of a bounded inverse for (P θ (h) − z) the restriction of π θ (P θ (h) − z) to u ∈ L 2 (R n x × R p y ) : π θ u = u . It is then elementary to verify that A θ z (h) is invertible, and its inverse A θ z (h) −1 is given by ( [3]):  is a compact operator on L 2 (R n x ) ⊕ L 2 (R n x ) , for θ and z complex small enough. K θ (z, h) depends analytically on z and lim z∈R,z→−∞ K θ (e −2θ z, h) = 0. By Theorem 1.2, we deduce that (I + K θ (z, h)) −1 is a z-meromorphic family for z in a complex neighborhood of 0. So it is the same for F θ (z) − z −1 = F θ − z (I + K θ (z, h)) −1 and (see [3]): We can also deduce by construction that the spectra of P θ (h) is discrete near 0 and z ∈ Σ(h) if and only if there exists θ ∈ C, Imθ > 0, |θ| small enough such that z ∈ σ disc F θ (z) . Conflicts of Interest: The author(s) declare that there are no conflicts of interest regarding the publication of this paper.