# Semiclassical limit to the Vlasov equation with inverse power law potentials

## TL;DR

This paper rigorously derives the classical Vlasov equation as the semiclassical limit of quantum fermionic systems with singular inverse power law potentials, providing explicit convergence rates under certain initial conditions.

## Contribution

It establishes strong convergence of quantum dynamics to the Vlasov equation for singular potentials and quantifies the convergence rate, extending previous results to more singular interactions.

## Key findings

- Strong convergence in trace and Hilbert-Schmidt norms
- Explicit bounds on convergence rate for $|x|^{-eta}$ potentials
- Applicable to initial states with specific integrability and regularity

## Abstract

We consider mixed quasi-free states describing $N$ fermions in the mean-field limit. In this regime, the time evolution is governed by the nonlinear Hartree equation. In the large $N$ limit, we study the convergence towards the classical Vlasov equation. Under integrability and regularity assumptions on the initial state, we prove strong convergence in trace and Hilbert-Schmidt norm and provide explicit bounds on the convergence rate for a class of singular potentials of the form $V(x)=|x|^{-\alpha}$, for $\alpha\in(0,1/2)$.