Strong phase-space semiclassical asymptotics

Abstract

Wigner and Husimi transforms have long been used for the phase-space reformulation of Schrödinger-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.

References in corpus (4)

• Coarse-scale representations and smoothed Wigner transforms
• On the time evolution of Wigner measures for Schrodinger equations
• Exact equations for smoothed Wigner transforms and homogenization of wave propagation
• Smoothed Wigner transforms in the numerical simulation of semiclassical (high-frequency) wave propagation

Cited by in corpus (3)

• Strong and weak semiclassical limits for some rough Hamiltonians
• Convergence rate in Wasserstein distance and semiclassical limit for the defocusing logarithmic Schr{ö}dinger equation
• On the selection of the classical limit for potentials with BV derivatives