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

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 $N=\sum_{q=1}^{p}N_{q}$.