The magnetic Liouville equation as a semi-classical limit

TL;DR

This paper demonstrates how the magnetic Liouville equation emerges as a semi-classical limit of the Heisenberg equation with a magnetic field, providing convergence results and uniform estimates in a 2D setting.

Contribution

It introduces a magnetic variant of semi-classical limits, establishing convergence and uniform estimates for the magnetic Liouville equation from the Heisenberg equation.

Findings

Convergence of the Heisenberg equation to the magnetic Liouville equation in semi-classical limit.

Uniform estimates in  and  for a specific 2D magnetic vector potential.

An application of an observation inequality for the Heisenberg equation with magnetic potential.

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.