Bose-Einstein condensation in an external potential at zero temperature: General theory

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Abstract
Bose-Einstein condensation is described in terms of the condensate wave function and the pair-excitation function, the latter being responsible for the existence of phonons. This minimal description in terms of these two functions is generalized to the case with an external potential. For a dilute gas with short-range pairwise repulsive interaction and at very low temperatures when the Bose-Einstein condensation is nearly complete, a partial differential equation is obtained for the condensate wave function and an integro-differential equation for the pair excitation. Experimentally, the external potential is used to trap the atoms, i.e., to keep them together. Since the trap is of macroscopic dimensions, the resulting external potential is often slowly varying. In these cases and when the condensate is in the lowest state, the partial differential equation for the condensate wave function and the integro-differential equation for the pair excitation are solved approximately for the case of a time-independent trap.