Semiclassical limit for mixed states with singular and rough potentials

Alessio Figalli (1) , Marilena Ligabo (2) , Thierry Paul (3)

1 Department of Mathematics and Statistics [Texas Tech]
2 Dipartimento di Matematica
3 CMLS - Centre de Mathématiques Laurent Schwartz

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).

Mots clés: Quantum Mechanics; Semiclassical approximation; Partial Differential Equations

Domaines: Equations aux dérivées partielles [math.AP]

Alessio Figalli, Marilena Ligabo, Thierry Paul. Semiclassical limit for mixed states with singular and rough potentials. Indiana University Mathematics Journal, 2012, 61 (1), pp.193-222. ⟨hal-00545715⟩