Derivation of the nonlinear Schr\"odinger equation from a many body Coulomb system

TL;DR

This paper proves that the dynamics of a large bosonic Coulomb system can be effectively described by the nonlinear Hartree equation, establishing a rigorous link between many-body quantum mechanics and mean-field models.

Contribution

It demonstrates the derivation of the nonlinear Schrödinger (Hartree) equation from a many-body Coulomb system with rigorous mathematical proof, including new a priori estimates.

Findings

Correlation functions factorize in the limit as N approaches infinity.

The limiting one-particle density matrix satisfies the nonlinear Hartree equation.

Established the uniqueness of the BBGKY hierarchy for the system.

Abstract

We consider the time evolution of N bosonic particles interacting via a mean field Coulomb potential. Suppose the initial state is a product wavefunction. We show that at any finite time the correlation functions factorize in the limit $N \to \infty$. Furthermore, the limiting one particle density matrix satisfies the nonlinear Hartree equation. The key ingredients are the uniqueness of the BBGKY hierarchy for the correlation functions and a new apriori estimate for the many-body Schr\"odinger equations.