Rate of Convergence Towards Semi-Relativistic Hartree Dynamics

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 -> a and G -> 0 while GN = lambda fixed. In the super-critical regime of large lambda, we introduce the regularized interaction where the cutoff vanishes as N -> a. We show that the difference between the many-body semi-relativistic Schrodinger 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.