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

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 ${\mathrm{\Psi }}_{N,0}={\phi }_0^{\otimes N}$ 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 $O(N^{-(1+4a)\mathrm{/}(3+2a)})$ for ${\phi }_0\in H^{(3\mathrm{/}2)+a}({\mathbb{R}}^3)$ where $0\le a<1\mathrm{/}2$ and $O(N^{-1})$ for $a>1$. Our proof is based on and extends the Fock space approach.