On the time dependence of the rate of convergence towards Hartree dynamics for interacting Bosons

We consider interacting $N$-Bosons in three dimensions. It is known that the difference between the many-body Schr\"odinger evolution in the mean-field regime and the corresponding Hartree dynamics is of order $1/N$. We investigate the time dependence of the difference. To have sub-exponential bound, we use the results of time decay estimate for small initial data. We also refine time dependent bound for singular potential using Strichartz estimate. We consider the interaction potential $V(x)$ of type $\lambda\exp(-\mu|x|)|x|^{-\gamma}$ for $\lambda\in\mathbb{R}$, $\mu\geq0$, and $0<\gamma<3/2$, which covers the Coulomb and Yukawa interaction.