# Rate of convergence towards equations of Hartree type for mixture condensates with factorized initial data

## TL;DR

This paper proves that for a multi-component bosonic system with Coulomb interactions, the many-body Schrödinger evolution converges to the Hartree equations at a rate of O(N^{-1}) in the mean-field regime.

## Contribution

It establishes a quantitative convergence rate of the many-body dynamics towards the Hartree equations for mixture condensates with factorized initial states.

## Key findings

- Convergence rate of O(N^{-1}) for the mean-field limit.
- Applicable to systems with Coulomb and generalized interactions.
- Initial data as a tensor product of factorized states.

## Abstract

We consider a system of $p$ components of bosons, each of which consists of $N_{1},N_{2},\dots,N_{p}$ particles, respectively. The bosons are in three dimensions with interactions via a generalized interaction potential which includes the Coulomb interaction. We set the initial condition to describe a mixture condensate, i.e., a tensor product of factorized states. We show that the difference between the many-body Schr\"odinger evolution in the mean-field regime and the corresponding $p$-particle dynamics due to a system of Hartree equation is $O(N^{-1})$ where $N=\sum_{q=1}^{p}N_{q}$.