Calculus of Variations and Geometric Measure Theory

A. Figalli - M. Ligabò - T. Paul

Semiclassical limit for mixed states with singular and rough potentials

created by figalli on 16 Sep 2011
modified on 05 Feb 2013


Accepted Paper

Inserted: 16 sep 2011
Last Updated: 5 feb 2013

Journal: Indiana Univ. Math. J.
Year: 2011


We consider the semiclassical limit for the Heisenberg-von Neumann equation with a potential which consists of the sum of a repulsive Coulomb potential, plus a Lipschitz potential whose gradient belongs to $BV$; this assumption on the potential guarantees the well posedness of the Liouville equation in the space of bounded integrable solutions. We find sufficient conditions on the initial data to ensure that the quantum dynamics converges to the classical one. More precisely, we consider the Husimi functions of the solution of the Heisenberg-von Neumann equation, and under suitable assumptions on the initial data we prove that they converge, as $\varepsilon \to 0$, to the unique bounded solution of the Liouville equation (locally uniformly in time).