Mean field evolution of fermions with Coulomb interaction

TL;DR

This paper proves that the evolution of weakly interacting fermions with Coulomb forces converges to Hartree-Fock dynamics under certain conditions, advancing understanding of many-body quantum systems.

Contribution

It establishes convergence of many-body fermion evolution to Hartree-Fock dynamics in a combined mean field and semiclassical regime, for Coulomb interactions.

Findings

Convergence proven for initial Slater determinant states.

Result holds under specific conditions on Hartree-Fock solutions.

Applicable to translation invariant initial data.

Abstract

We study the many body Schr\"odinger evolution of weakly coupled fermions interacting through a Coulomb potential. We are interested in a joint mean field and semiclassical scaling, that emerges naturally for initially confined particles. For initial data describing approximate Slater determinants, we prove convergence of the many-body evolution towards Hartree-Fock dynamics. Our result holds under a condition on the solution of the Hartree-Fock equation, that we can only show in a very special situation (translation invariant data, whose Hartree-Fock evolution is trivial), but that we expect to hold more generally.