Abstract view
SemiClassical Behavior of the Scattering Amplitude for Trapping Perturbations at Fixed Energy


Published:20040801
Printed: Aug 2004
Abstract
We study the semiclassical 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 shortrange 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 thenontrapping case.