Calculus of Variations and Geometric Measure Theory

L. Ambrosio - A. Figalli - G. Friesecke - J. Giannoulis - T. Paul

Semiclassical limit of quantum dynamics with rough potentials and well posedness of transport equations with measure initial data

created by ambrosio on 27 Jun 2010
modified by figalli on 08 Dec 2012


Accepted Paper

Inserted: 27 jun 2010
Last Updated: 8 dec 2012

Journal: Comm. Pure Appl. Math.
Year: 2011


We prove convergence of the Wigner transforms of solutions to the Schrodinger equation, in a semiclassical limit, to solutions to the Liouville equation. We are able to include in our convergence result rough or singular potentials (with Coulomb repulsive singularities) provided convergence is understood for ``almost all'' initial data. The rigorous statement involves a suitable extension of the DiPerna-Lions theory to the infinite-dimensional space of probability measure, where both the Wigner and the Liouville dynamics can be read. The paper is a continuation of previous work by Ambrosio, Friesecke and Giannoulis.

Tags: GeMeThNES
Keywords: Schrodinger equation, Wigner transform, Liouville equation, Semiclassical limit