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

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