Uniform in Time Convergence to Bose-Einstein Condensation for a Weakly Interacting Bose Gas with an External Potential

TL;DR

This paper proves that the evolution of a weakly interacting Bose gas with an external potential converges uniformly in time to a mean-field Hartree equation, with optimal error dependence on particle number.

Contribution

It establishes uniform-in-time convergence to Bose-Einstein condensation for a weakly interacting Bose gas with external potential, including intermediate regimes.

Findings

Uniform in time error bounds for the Hartree approximation

Optimal dependence of error on particle number

Extension to intermediate regimes between mean field and Gross-Pitaevskii

Abstract

We consider a gas of weakly interacting bosons in three dimensions subject to an external potential in the mean field regime. Assuming that the initial state of our system is a product state, we show that in the trace topology of one-body density matrices, the dynamics of the system can be described by the solution to the corresponding Hartree type equation. Using a dispersive estimate for the Hartree type equation, we obtain an error term that is uniform in time. Moreover, the dependence of the error term on the particle number is optimal. We also consider a class of intermediate regimes between the mean field regime and the Gross-Pitaevskii regime, where the error term is uniform in time but not optimal in the number of particles.