A Mixed-Norm Estimate of the Two-Particle Reduced Density Matrix of Many-Body Schrödinger Dynamics for Deriving the Vlasov Equation

Abstract

We re-examine the combined semi-classical and mean-field limit in the N-body fermionic Schrödinger 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 Chen et al. (J Stat Phys 182(2):1–41, http://arxiv.org/abs/1910.09892v4, 2021). 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 Chen et al. (J Stat Phys 182(2):1–41, http://arxiv.org/abs/1910.09892v4, 2021).