Semi-classical limit of quantum free energy minimizers for the gravitational Hartree equation

TL;DR

This paper investigates the transition from quantum to classical states in gravitational systems by analyzing how quantum free energy minimizers converge to classical ones as Planck's constant approaches zero, using various mathematical tools.

Contribution

It establishes the rigorous convergence of quantum free energy minimizers to classical minimizers in the semi-classical limit for the gravitational Hartree equation.

Findings

Quantum minimizers converge to classical minimizers as 

Convergence shown via potential functions, Husimi transform, and Tfoplitz quantization

Provides a mathematical bridge between quantum and classical gravitational models

Abstract

For the gravitational Vlasov-Poisson equation, Guo and Rein 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. In this paper, we are concerned with the correspondence between quantum and classical isotropic states. Precisely, we prove that as the Planck constant $\hbar$ 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 Husimi transform and the T\"oplitz quantization.