Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

TL;DR

This paper analyzes the convergence rate of the semi-relativistic Hartree dynamics for a large system of gravitating bosons, establishing an optimal bound uniform across sub-critical and super-critical regimes.

Contribution

It introduces a regularized interaction in the super-critical regime and proves an optimal convergence rate of order N^{-1} between many-body and Hartree dynamics.

Findings

Convergence rate of order N^{-1} established

Uniform bound across sub-critical and super-critical regimes

Regularized interaction handles super-critical regime effectively

Abstract

We consider the semi-relativistic system of $N$ gravitating Bosons with gravitation constant $G$. The time evolution of the system is described by the relativistic dispersion law, and we assume the mean-field scaling of the interaction where $N \to \infty$ and $G \to 0$ while $GN = \lambda$ fixed. In the super-critical regime of large $\lambda$, we introduce the regularized interaction where the cutoff vanishes as $N \to \infty$. We show that the difference between the many-body semi-relativistic Schr\"{o}dinger dynamics and the corresponding semi-relativistic Hartree dynamics is at most of order $N^{-1}$ for all $\lambda$, i.e., the result covers the sub-critical regime and the super-critical regime. The $N$ dependence of the bound is optimal.