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

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.

arXiv comment$\beta >0$ case part is removed

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