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We investigate the long-range phase coherence of homogeneous and trapped Bosegases as a function of the geometry of the trap, the temperature, and themean-field interactions in the weakly interacting limit. We explicitly takeinto account the (quasi)condensate depletion due to quantum and thermalfluctuations, i.e., we include the effects of both phase and densityfluctuations. In particular, we determine the phase diagram of the gas bycalculating the off-diagonal one-particle density matrix and discuss thevarious crossovers that occur in this phase diagram and the feasibility oftheir experimental observation in trapped Bose gases.Comment: One figure added, typos corrected, refernces adde

Dimensional and temperature crossover in trapped Bose gases