On the Mean-Field Limit of Bosons with Coulomb Two-Body Interaction

TL;DR

This paper introduces a straightforward method to prove the convergence of quantum Bose gas dynamics to the Hartree equation in the mean-field limit, including for singular Coulomb interactions, emphasizing a semi-classical perspective.

Contribution

The paper presents a new simple approach to establish the mean-field limit for Bose gases with Coulomb interactions, avoiding coherent states and highlighting a semi-classical framework.

Findings

Proves convergence of quantum dynamics to Hartree dynamics for Coulomb potentials.

Applicable to a class of singular interaction potentials.

Highlights the semi-classical nature of the mean-field limit.

Abstract

In the mean-field limit the dynamics of a quantum Bose gas is described by a Hartree equation. We present a simple method for proving the convergence of the microscopic quantum dynamics to the Hartree dynamics when the number of particles becomes large and the strength of the two-body potential tends to 0 like the inverse of the particle number. Our method is applicable for a class of singular interaction potentials including the Coulomb potential. We prove and state our main result for the Heisenberg-picture dynamics of "observables", thus avoiding the use of coherent states. Our formulation shows that the mean-field limit is a "semi-classical" limit.