Semiclassical limit for mixed states with singular and rough potentials

TL;DR

This paper investigates the semiclassical limit of quantum dynamics described by the Heisenberg-von Neumann equation with singular and rough potentials, showing convergence to classical Liouville dynamics under certain initial conditions.

Contribution

It establishes conditions under which quantum states with singular potentials converge to classical phase space distributions in the semiclassical limit.

Findings

Quantum dynamics converge to classical Liouville solutions as epsilon approaches zero.

Convergence holds for initial data satisfying specific regularity conditions.

The results apply to potentials with Coulomb singularities and Lipschitz continuity.

Abstract

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 $\e \to 0$, to the unique bounded solution of the Liouville equation (locally uniformly in time).