Inserted: 27 jun 2010
Last Updated: 8 dec 2012
Journal: Comm. Pure Appl. Math.
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.
Keywords: Schrodinger equation, Wigner transform, Liouville equation, Semiclassical limit