Adiabatic Evolution and Shape Resonances

"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 Schrodinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter with ln , where h denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length N with an error O(). HereN 0 is arbitrary"--