The magnetic Liouville equation as a semi-classical limit

Abstract

The Liouville equation with non-constant magnetic field is obtained as a limit in the Planck constant \hbar of the Heisenberg equation with the same magnetic field. The convergence is with respect to an appropriate semi-classical pseudo distance, and consequently with respect to the Monge-Kantorovich distance. Uniform estimates both in \epsilon and \hbar are proved for the specific 2D case of a magnetic vector potential of the form \frac {1} {\epsilon}x^{\bot}. As an application, an observation inequality for the Heisenberg equation with a magnetic vector potential is obtained. These results are a magnetic variant of the works [7] and [8] respectively.