Semiclassical asymptotics of the residues of the scattering matrix for shape resonances

Article type: Research Article

Authors: Lahmar‐Benbernou, Amina | Martinez, André^;

Affiliations: Département de Mathématiques, ENS – SF de Mostaganem, P.O. Box 227, 27000 Mostaganem, Algeria | Dipartimento di Matematica, Università di Bologna, Piazza di porta San Donato, 5, 40127 Bologna, Italy

Note: [] Corresponding author. E‐mail: martinez@dm.unibo.it.

Abstract: The aim of this study is to give complete semiclassical asymptotics of the residues {\rm Res}[S(\lambda,\omega,\omega^\prime),\rho] at some pole \rho of the distributional kernel of the scattering matrix S(\lambda) corresponding to a semiclassical two‐body Schrödinger operator P=-h^2\Delta + V, and considered as a meromorphic operator‐valued function with respect to the energy \lambda. We do it in the case where the pole \rho considered is a shape resonance of P. This is a continuation of A. Benbernou, Estimation des residus de la matrice de diffusion associés a dès résonances de forme I (to appear in Ann. Inst. H. Poincaré), where an extra geometrical condition was assumed (namely the absence of caustics near the energy level {\rm Re}\ \rho). Here we drop this assumption by using an FBI transform which permits to work in the complexified phase space. Then we show that some semiclassical WKB expansions are global, and this allows us to find out estimates for the residue of the type {\rm O}(h^N{\rm e}^{-2S_0/h}), where S_0 is the Agmon width of the potential barrier, and N may be arbitrarily large depending on an explicit geometrical location of the incoming and outgoing waves \omega and \omega' one consider. Full asymptotic expansions are obtained under some additional generic geometric assumption on the potential V.

Journal: Asymptotic Analysis, vol. 20, no. 1, pp. 13-38, 1999