×

Semiclassical asymptotics of the residues of the scattering matrix for shape resonances. (English) Zbl 0931.35119

Summary: The aim of this study is to give complete semiclassical asymptotics of the residues \(\text{Res}[S(\lambda,\omega,\omega'),\) \(\rho]\) 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 \(\text{Re} \rho)\). 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 \(O(h^N e^{-2S_0/h})\), 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 \(V\).

MSC:

35P25 Scattering theory for PDEs
35J10 Schrödinger operator, Schrödinger equation
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory