Adiabatic evolution and shape resonances

TL;DR

This paper studies adiabatic evolution in semi-classical Schrödinger operators with time-dependent potentials, providing approximations of solutions over long times with controllable errors, relevant for understanding shape resonances.

Contribution

It establishes a method to choose the adiabatic parameter in relation to the semi-classical parameter, enabling precise long-time approximations of solutions.

Findings

Adiabatic approximations valid over long time intervals.

Error estimates of order ${ m O}(oldsymbol{ m ext{error}})}$ for solutions.

Relation between adiabatic parameter and semi-classical parameter.

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 Schr\"odinger operator with a time dependent potential with a well in an island. In particular, we show that we can choose the adiabatic parameter $\varepsilon $ with $\ln\varepsilon \asymp -1/h$, where $h$ denotes the semi-classical parameter, and get adiabatic approximations of exact solutions over a time interval of length $\varepsilon ^{-N}$ with an error ${\cal O}(\varepsilon ^N)$. Here $N>0$ is arbitrary.