Egorov's theorem in the Weyl--H\"ormander calculus

TL;DR

This paper extends Egorov's theorem within the Weyl--H"ormander calculus, providing a broad framework for understanding evolution propagators in phase space with applications to quantum mechanics.

Contribution

It establishes a general Egorov's theorem in the Weyl--H"ormander setting, including quantification of Ehrenfest time and detailed symbol analysis.

Findings

Proves a generalized Egorov's theorem for various evolution equations.

Quantifies Ehrenfest time in the Weyl--H"ormander framework.

Describes the full symbol of conjugated operators.

Abstract

We prove a general version of Egorov's theorem for evolution propagators in the Euclidean space, in the Weyl--H\"ormander framework of metrics on the phase space. Mild assumptions on the Hamiltonian allow for a wide range of applications that we describe in the paper, including Schr\"odinger, wave and transport evolutions. We also quantify an Ehrenfest time and describe the full symbol of the conjugated operator. Our main result is a consequence of a stronger theorem on the propagation of quantum partitions of unity.