Semi-classical behavior of the scattering amplitude for trapping perturbations (Q1600208)

We study the semi-classical behavior when \(h\to 0\), of the scattering amplitude \(f(\theta, \omega, \lambda, h)\) associated to the Schrödinger operator \(P(h)=-\frac{1}{2}h^2\Delta+V(x)\) in \(x\in\mathbb{R}^n\) for short range perturbations \(V(x)\). We show that if we modify the potential \(V(x)\) in a domain contained in \(\{x\;:\;V(x)>\lambda\}\), the scattering amplitude \(f(\theta, \omega, \lambda, h)\) is changed by a term of order \(\mathcal{O}(h^\infty)\). Moreover, under an additional escape assumption, we get an asymptotics of the scattering amplitude. See also the detailed version in Can. J. Math. 56, No. 4, 794--824 (2004; Zbl 1084.35067).