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 as- sumptions on the initial data we prove that they converge, as ε → 0, to the unique bounded solution of the Liouville equation (locally uniformly in time).
Semiclassical Limit for Mixed States with Singular and Rough Potentials
LIGABO', MARILENA;
2012-01-01
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 as- sumptions on the initial data we prove that they converge, as ε → 0, to the unique bounded solution of the Liouville equation (locally uniformly in time).I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
