Strong phase-space semiclassical asymptotics

TL;DR

This paper advances the understanding of semiclassical limits by establishing strong topology approximations of Wigner functions with solutions of the Liouville equation, emphasizing the importance of potential regularity.

Contribution

It provides new results on strong convergence of Wigner functions in $L^2$ and Sobolev norms, extending semiclassical analysis beyond weak topologies and considering potential regularity effects.

Findings

Strong convergence of Wigner functions in $L^2$ and Sobolev norms.

Approximation valid up to $O(log(1/\epsilon))$ time-scales.

Highlights role of potential regularity in semiclassical limits.

Abstract

Wigner and Husimi transforms have long been used for the phase-space reformulation of Schr\"odinger-type equations, and the study of the corresponding semiclassical limits. Most of the existing results provide approximations in appropriate weak topologies. In this work we are concerned with semiclassical limits in the strong topology, i.e. approximation of Wigner functions by solutions of the Liouville equation in $L^2$ and Sobolev norms. The results obtained improve the state of the art, and highlight the role of potential regularity, especially through the regularity of the Wigner equation. It must be mentioned that the strong convergence can be shown up to $O(log \frac{1}\epsilon)$ time-scales, which is well known to be, in general, the limit of validity of semiclassical asymptotics.