Mean-Field Limit of Quantum Bose Gases and Nonlinear 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.

27 pages, 2 figures

References in corpus (1)

• Asymptotic stability of N-soliton states of NLS

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• The fractional in time Schrödinger equation with a Hartree perturbation
• Semi-classical states for the nonlinear Choquard equations: existence, multiplicity and concentration at a potential well