98-490 Lahmar-Benbernou A., Martinez A.

Semiclassical Asymptotics of the Residues of the Scattering Matrix for Shape Resonances (83K, LATeX 2e) Jul 6, 98

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Abstract. The aim of this study is to give complete semiclassical asymptotics of the residues 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 Schr\"{o}dinger 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 \cite{Be2} where an extra geometrical condition was assumed (namely the absence of caustics near the energy level 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 ${\cal O}(h^N e^{-2S_0/h})$ 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 $V$.

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