The classical limit of the time dependent Hartree-Fock equation. I. The Weyl symbol of the solution

TL;DR

This paper investigates the evolution of the Weyl symbol in the time dependent Hartree-Fock equation, proving its integrability preservation over time and providing an asymptotic expansion in phase space.

Contribution

It establishes the preservation of integrability of the Weyl symbol over time and derives an asymptotic expansion for its solution.

Findings

Weyl symbol remains integrable for all times

Provides an asymptotic expansion in L1 sense

Shows stability of phase space properties over time

Abstract

We study the time evolution of the Weyl symbol of a solution of the time dependent Hartree Fock equation, assuming that for t=0, it has a Weyl symbol which is integrable in the phase space, such as all its derivatives. We prove that the solution has the same property for all t, and we give an asymptotic expansion, in L1 sense, of this Weyl symbol.