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

Published:2004-08-01
Printed: Aug 2004
• Laurent Michel
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## Abstract

We study the semi-classical behavior as $h\rightarrow 0$ of the scattering amplitude $f(\theta,\omega,\lambda,h)$ associated to a Schr\"odinger 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 $\{x:V(x)>\lambda\}$, the scattering amplitude $f(\theta,\omega,\lambda,h)$ changes by a term of order $\O(h^{\infty})$. 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 thenon-trapping case.
 MSC Classifications: 35P25 - Scattering theory [See also 47A40] 35B34 - Resonances 35B40 - Asymptotic behavior of solutions

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