On the time evolution of Wigner measures for Schrodinger equations

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.

Survey, 26 pages

Cited by in corpus (9)

• Semiclassical propagation of Wigner functions
• Coarse-scale representations and smoothed Wigner transforms
• Strong phase-space semiclassical asymptotics
• Wavepackets in inhomogeneous periodic media: effective particle-field dynamics and Berry curvature
• Localisation principle for one-scale H-measures
• An asymptotic preserving scheme based on a new formulation for NLS in the semiclassical limit
• Semiclassical evolution of quantum wave packets on the torus beyond the Ehrenfest time in terms of Husimi distributions
• Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schr{ö}dinger equation
• Semi-classical limit of the massive Klein-Gordon-Maxwell system toward the relativistic Euler-Maxwell system via an adapted modulated energy method