On the time evolution of Wigner measures for Schrodinger equations

TL;DR

This survey discusses the limitations of Wigner measures in analyzing the high-frequency limits of Schrödinger equations, highlighting cases where they fail to capture essential effects or lead to ill-posed propagation problems.

Contribution

It provides a comprehensive review of known limitations of Wigner measures in semi-classical analysis of Schrödinger equations, including cases where refined measures also fail.

Findings

Wigner measures may not capture non-negligible pointwise effects.

Propagation of Wigner measures can be ill-posed.

Refined Wigner measures like two-scale measures can also fail in some situations.

Abstract

In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.