Rate of Convergence towards Hartree Dynamics with Singular Interaction Potential

TL;DR

This paper proves that for a system of bosons with possibly singular two-body interactions, the many-body Schrödinger evolution converges to the Hartree dynamics at an optimal rate of 1/N, given H^1 initial data.

Contribution

It establishes an optimal convergence rate of 1/N for the mean-field approximation with singular potentials, extending previous results to more general interactions.

Findings

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

Applicable to singular potentials beyond Coulomb.

Optimal N-dependence of the convergence bound.

Abstract

We consider a system of $N$-Bosons with a two-body interaction potential $V \in L^2(\mathbb{R}^3)+L^\infty(\mathbb{R}^3)$, possibly singular than the Coulomb interaction. We show that, with $H^1(\mathbb{R}^3)$ initial data, the difference between the many-body Schr\"odinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order $1/N$, for any fixed time. The $N$-dependence of the bound is optimal.