Mean-field evolution of fermions with singular interaction

TL;DR

This paper proves that the many-body Schrödinger dynamics of fermions with singular inverse power law interactions converges to the time-dependent Hartree-Fock equation, highlighting how potential singularity affects initial data regularity.

Contribution

It extends previous results to include potentials with singularity parameter lpha in (0,1], demonstrating convergence in the mean-field regime for these more singular interactions.

Findings

Convergence of many-body Schrödinger solutions to Hartree-Fock equations.

Dependence of initial data regularity on potential singularity.

Extension of previous lpha=1 case to lpha in (0,1]

Abstract

We consider a system of N fermions in the mean-field regime interacting though an inverse power law potential $V(x)=1/|x|^{\alpha}$, for $\alpha\in(0,1]$. We prove the convergence of a solution of the many-body Schr\"{o}dinger equation to a solution of the time-dependent Hartree-Fock equation in the sense of reduced density matrices. We stress the dependence on the singularity of the potential in the regularity of the initial data. The proof is an adaptation of [22], where the case $\alpha=1$ is treated.