Rate of convergence towards mean-field evolution for weakly interacting bosons with singular three-body interactions

Abstract

In this paper, we investigate the dynamics of a system of $N$ weakly interacting bosons with singular three-body interactions in three dimensions. By assuming factorized initial data $\Psi_{N,0}=\varphi_{0}^{\otimes N}$ and triple collisions, we prove that in the many-particle limit, its mean-field approximation converges to quintic Hartree dynamics. Moreover, we prove that the rate of convergence towards the mean-field quintic Hartree evolution is of $O(N^{-(1+4a)/(3+2a)})$ for $\varphi_{0}\in H^{(3/2)+a}(\mathbb{R}^{3})$ where $0\leq a<1/2$ and $O(N^{-1})$ for $a>1$. Our proof is based on and extends the Fock space approach.