Semi-Classical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy

Michel, Laurent

Canadian journal of mathematics, Tome 56 (2004) no. 4, pp. 794-824

Résumé

We study the semi-classical behavior as $h\,\to \,0$ of the scattering amplitude $f(\theta ,\,\omega ,\,\lambda ,\,h)$ associated to a Schrödinger operator $P(h)\,=\,-\,\frac{1}{2}{{h}^{2}}\Delta \,+\,V\,(x)$ with short-range trapping perturbations. First we realize a spatial localization in the general case and we deduce a bound of the scattering amplitude on the real line. Under an additional assumption on the resonances, we show that if we modify the potential $V(x)$ in a domain lying behind the barrier $\left\{ x\,:\,V(x)\,>\,\lambda\right\}$ , the scattering amplitude $f(\theta ,\,\omega ,\,\lambda ,\,h)$ changes by a term of order $\mathcal{O}({{h}^{\infty }})$ . Under an escape assumption on the classical trajectories incoming with fixed direction $\omega $ , we obtain an asymptotic development of $f(\theta ,\,\omega ,\,\lambda ,\,h)$ similar to the one established in the non-trapping case.

Détail

DOI : 10.4153/CJM-2004-036-2

Mots-clés : 35P25, 35B34, 35B40

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