Propagation of Moments and Semiclassical Limit from Hartree to Vlasov Equation

TL;DR

This paper establishes a quantitative connection between the Hartree and Vlasov equations in the semiclassical limit, handling singular interactions like Coulomb potential, and demonstrates the propagation of moments and norms ensuring uniform boundedness.

Contribution

It provides a rigorous, quantitative proof of the semiclassical limit from Hartree to Vlasov equations with singular interactions, including Coulomb potential.

Findings

Proves propagation of velocity moments and weighted Schatten norms.

Establishes uniform boundedness of particle space density in Planck constant.

Quantifies the semiclassical limit with singular interactions.

Abstract

In this paper, we prove a quantitative version of the semiclassical limit from the Hartree to the Vlasov equation with singular interaction, including the Coulomb potential. To reach this objective, we also prove the propagation of velocity moments and weighted Schatten norms which implies the boundedness of the space density of particles uniformly in the Planck constant.