Semi-classical behavior of the scattering amplitude for trapping perturbations at fixed energy

Abstract. We study the semi-classical behavior as h → 0 of the scattering amplitude f (θ, ω, λ, h) associated to a Schrödinger operator P(h) = − 1 2 h2 ∆ + 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)> λ}, the scattering amplitude f (θ, ω,λ, h) changes by a term of order O(h ∞). Under an escape assumption on the classical trajectories incoming with fixed direction ω, we obtain an asymptotic development of f (θ, ω, λ, h) similar to the one established in the non-trapping case