Smoothing effect and quantum-classical correspondence for the Schr{\"o}dinger equation with confining potential

TL;DR

This paper demonstrates that for the Schrödinger equation with confining potentials, the smoothing effect is equivalent to classical flow escape rate estimates, linking quantum regularity improvements to classical dynamics.

Contribution

It establishes an equivalence between the smoothing effect and classical escape rate estimates for Schrödinger equations with sub-quadratic confining potentials.

Findings

Smoothing effect is equivalent to classical escape rate estimates.

Proof relies on an Egorov theorem from prior work.

Results connect quantum regularity with classical flow dynamics.

Abstract

The smoothing effect states that solutions to the Schr{\"o}dinger equation in the Euclidean space have, for almost-every time, a local-in-space improved regularity (gain of half a derivative in Sobolev spaces). In this note, we show that, for the Schr{\"o}dinger equation with a sub-quadratic confining potential, the smoothing effect is equivalent to an escape rate estimate on the associated classical flow. The proof relies on an Egorov theorem proved in~\cite{P:24Egorovinprep}.