Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation

TL;DR

This paper explores the derivation, stability, and classical limit of the Hartree equation in the mean-field regime of large bosonic systems, emphasizing focusing nonlinearities and their implications in quantum and classical dynamics.

Contribution

It provides rigorous derivations of the Hartree equation from quantum many-body theory and analyzes the existence, stability, and classical limit of solutions with focusing nonlinearities.

Findings

Rigorous derivation of the Hartree equation from quantum bosonic systems

Existence and stability results for Hartree solitons

Analysis of the point-particle (Newtonian) limit of the Hartree equation

Abstract

We discuss the Hartree equation arising in the mean-field limit of large systems of bosons and explain its importance within the class of nonlinear Schroedinger equations. Of special interest to us is the Hartree equation with focusing nonlinearity (attractive two-body interactions). Rigorous results for the Hartree equation are presented concerning: 1) its derivation from the quantum theory of large systems of bosons, 2) existence and stability of Hartree solitons, and 3) its point-particle (Newtonian) limit. Some open problems are described.