Semiclassical resolvent estimates and resonances free regions for Schrödinger operators with matrix-valued potentials

Marouane Assal, Pontificia Universidad Católica de Chile

Abstract

For semiclassical Schrödinger operators with scalar long-range potentials, a classical result due to Burq ensures that the resolvent norm grows exponentially in the inverse of the semiclassical parameter, and grows linearly near infinity. This result holds without any assumption on the classical dynamics and it implies in particular the absence of resonances exponentially close to the real axis. In this talk, I will present a generalization of these results for semiclassical Schrödinger operators with matrix-valued long-range potentials without any assumption on the eigenvalues crossings. In particular, I will focus on an elementary approach introduced by Datchev based on an explicit global Carleman estimate. If time is enough, I will also present an application of these results to the study of the scattering amplitude.