# Global Semiclassical Limit from Hartree to Vlasov Equation for Concentrated Initial Data

## TL;DR

This paper establishes a rigorous, quantitative connection between the quantum Hartree equation and the classical Vlasov equation in higher dimensions, including Coulomb interactions, for highly concentrated initial states.

## Contribution

It provides the first global-in-time semiclassical limit results for singular potentials like Coulomb, with explicit bounds and conditions on initial data concentration.

## Key findings

- Quantitative bounds on moments of solutions
- Global-in-time convergence results
- Analysis of dispersion effects on spatial density

## Abstract

We prove a quantitative and global in time semiclassical limit from the Hartree to the Vlasov equation in the case of a singular interaction potential in dimension $d\geq 3$, including the case of a Coulomb singularity in dimension $d=3$. This result holds for initial data concentrated enough in the sense that some space moments are initially sufficiently small. As an intermediate result, we also obtain quantitative semiclassical bounds on the space and velocity moments of even order and the asymptotic behavior of the spatial density due to dispersion effects.