The Bogoliubov inequality and the nature of Bose-Einstein condensates for interacting atoms in spatial dimensions D le 2

We consider the restriction placed by the Bogoliubov inequality on the nature of the Bose-Einstein condensates (BECs) for interacting atoms in a spatial dimension D </- 2 and in the presence of an external arbitrary potential, which may be a confining "box", a periodic, or a disordered potential. The atom-atom interaction gives rise to a (gauge invariance) symmetry-breaking term that places further restrictions on BECs in the form of a consistency proviso. The necessary condition for the existence of a BEC in D </- 2 in all cases is macroscopic occupation of many single-particle momenta states with the origin a limit point (or accumulation point) of condensates. It is shown that the nature of BECs for noninteracting atoms in a disordered potential is precisely the same as that of BECs for interacting atoms in the absence of an external potential.