Semiclassical Limit to the Vlasov Equation with Inverse Power Law Potentials

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|-α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${V(x)=|x|^{-\alpha}}$$\end{document} , for α∈(0,1/2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\alpha\in(0,1/2)}$$\end{document} .