Mean-Field Dynamics: Singular Potentials and Rate of Convergence

TL;DR

This paper extends the analysis of mean-field dynamics for bosonic systems by including singular potentials and deriving convergence rates, broadening the applicability of the Hartree equation in quantum many-body physics.

Contribution

It introduces a nonperturbative method to handle singular potentials and provides explicit bounds on the convergence rate to the Hartree dynamics.

Findings

Extended mean-field results to include singular potentials.

Derived explicit convergence rate bounds.

Applicable to strong, time-dependent external potentials.

Abstract

We consider the time evolution of a system of $N$ identical bosons whose interaction potential is rescaled by $N^{-1}$. We choose the initial wave function to describe a condensate in which all particles are in the same one-particle state. It is well known that in the mean-field limit $N \to \infty$ the quantum $N$-body dynamics is governed by the nonlinear Hartree equation. Using a nonperturbative method, we extend previous results on the mean-field limit in two directions. First, we allow a large class of singular interaction potentials as well as strong, possibly time-dependent external potentials. Second, we derive bounds on the rate of convergence of the quantum $N$-body dynamics to the Hartree dynamics.