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

TL;DR

This paper analyzes how the difference between many-body Schrödinger evolution and Hartree dynamics for interacting bosons depends on time, providing refined bounds for specific potentials like Coulomb and Yukawa.

Contribution

It introduces time-dependent bounds on the convergence rate for bosonic systems with singular potentials, improving understanding of the dynamics over time.

Findings

Established sub-exponential bounds using decay estimates.

Refined bounds for singular potentials with Strichartz estimates.

Applicable to Coulomb and Yukawa interactions.

Abstract

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.