THE MAGNETIC LIOUVILLE EQUATION AS A SEMICLASSICAL LIMIT

The Liouville equation with nonconstant magnetic field is obtained as a limit in the Planck constant h of the von Neumann equation with the same magnetic field. The convergence is with respect to an appropriate semiclassical pseudodistance, and consequently with respect to the Monge-Kantorovich distance. Uniform estimates both in \epsilon and h are proved for the specific 2D case of a magnetic vector potential of the form 1\epsilon x\bot. As an application, an observation inequality for the von Neumann equation with a magnetic vector potential is obtained. These results are a magnetic variant of the works [F. Golse and T. Paul, Arch. Ration. Mech. Anal., 223 (2017), pp. 57--94] and [F. Golse and T. Paul, Math. Models Methods Appl. Sci., 32 (2022), pp. 941--963], respectively.