Adiabatic evolution and shape resonances

Abstract

Motivated by a problem of one mode approximation for a non-linear evolution with charge accumulation in potential wells, we consider a general linear adiabatic evolution problem for a semi-classical Schrödinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter $\epsilon$ with $\ln \epsilon \asymp -1/h$, where $h$ denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length ${\epsilon }^{-N}$ with an error $\mathcal{O}({\epsilon }^N)$. Here $N>0$ is arbitrary.