A mixed-norm estimate of the two-particle reduced density matrix of many-body Schr\"odinger dynamics for deriving the Vlasov equation

TL;DR

This paper improves estimates in the semi-classical and mean-field limit analysis of many-body fermionic Schr"odinger equations, using a mixed-norm approach to better understand the two-particle reduced density matrix and its role in deriving the Vlasov equation.

Contribution

It introduces a novel mixed-norm estimate for the two-particle reduced density matrix, enhancing the understanding of residue estimates in the semi-classical and mean-field limits.

Findings

Improved bounds for kinetic and mean-field residues.

Smaller mean-field residue estimate compared to previous work.

Refined oscillation estimates in residual term analysis.

Abstract

We re-examine the combined semi-classical and mean-field limit in the $N$-body fermionic Schr\"odinger equation with pure state initial data using the Husimi measure framework. The Husimi measure equation involves three residue types: kinetic, semiclassical, and mean-field. The main result of this paper is to provide better estimates for the kinetic and mean-field residue than those in \cite{Chen2021JSP}. Especially, the estimate for the mean-field residue is shown to be smaller than the semiclassical residue by a mixed-norm estimate of the two-particle reduced density matrix factorization. Our analysis also updates the oscillation estimate parts in the residual term estimates appeared in \cite{Chen2021JSP}.