Semiclassical Dynamics with Exponentially Small Error Estimates

TL;DR

This paper develops approximate solutions to the time-dependent Schrödinger equation for small , achieving exponentially small error bounds under certain conditions, advancing semiclassical analysis techniques.

Contribution

It constructs highly accurate semiclassical solutions with exponentially small errors, extending previous methods to longer times and more restrictive conditions.

Findings

Errors are bounded by C exp(-/) for small  and fixed times.

Under stronger conditions, errors are bounded by C' exp(-^) for times up to || log().

Provides rigorous error estimates for semiclassical approximations in quantum dynamics.

Abstract

We construct approximate solutions to the time--dependent Schr\"odinger equation $i \hbar (\partial \psi)/(\partial t) = - (\hbar^2)/2 \Delta \psi + V \psi$ for small values of $\hbar$. If $V$ satisfies appropriate analyticity and growth hypotheses and $|t|\le T$, these solutions agree with exact solutions up to errors whose norms are bounded by $C \exp{-\gamma/\hbar}$, for some $C$ and $\gamma>0$. Under more restrictive hypotheses, we prove that for sufficiently small $T', |t|\le T' |\log(\hbar)|$ implies the norms of the errors are bounded by $C' \exp{-\gamma'/\hbar^{\sigma}}$, for some $C', \gamma'>0$, and $\sigma>0$.