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

We re-examine the combined semi-classical and mean-field limit in the N -body fermionic Schrodinger 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).