Semi-classical Limit of Quantum Free Energy Minimizers for the Gravitational Hartree Equation

For the gravitational Vlasov–Poisson equation, Guo and Rein (Arch Rational Mech Anal 147(3):225–243, 1999) constructed a class of classical isotropic states as minimizers of free energies (or energy-Casimir functionals) under mass constraints. For the quantum counterpart, that is, the gravitational Hartree equation, isotropic states are constructed as free energy minimizers by Aki, Dolbeault and Sparber (Ann Henri Poincaré 12(6):1055–1079, 2011). In this paper, we are concerned with the correspondence between quantum and classical isotropic states. More precisely, we prove that as the Planck constant ħ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\hbar $$\end{document} goes to zero, free energy minimizers for the Hartree equation converge to those for the Vlasov–Poisson equation in terms of potential functions as well as via the Wigner transform and the Töplitz quantization.