Rate of Convergence Towards Hartree Dynamics

TL;DR

This paper proves that the mean-field approximation for a large system of interacting bosons converges to the Hartree dynamics at a rate of 1/N, even with Coulomb-like singular potentials, and this rate is proven to be optimal.

Contribution

It establishes an optimal convergence rate of 1/N for the mean-field approximation to Hartree dynamics in systems with singular potentials.

Findings

Convergence rate of 1/N between many-body Schrödinger evolution and Hartree dynamics.

Optimality of the convergence rate with respect to N.

Applicability to systems with Coulomb-type singularities.

Abstract

We consider a system of N bosons interacting through a two-body potential with, possibly, Coulomb-type singularities. We show that the difference between the many-body Schr\"odinger evolution in the mean-field regime and the effective nonlinear Hartree dynamics is at most of the order 1/N, for any fixed time. The N-dependence of the bound is optimal.