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We consider the restriction placed by the Bogoliubov inequality on the natureof the Bose-Einstein condensates (BECs) for interacting atoms in a spatialdimension D </- 2 and in the presence of an external arbitrary potential, whichmay be a confining "box", a periodic, or a disordered potential. The atom-atominteraction gives rise to a (gauge invariance) symmetry-breaking term thatplaces further restrictions on BECs in the form of a consistency proviso. Thenecessary condition for the existence of a BEC in D </- 2 in all cases ismacroscopic occupation of many single-particle momenta states with the origin alimit point (or accumulation point) of condensates. It is shown that the natureof BECs for noninteracting atoms in a disordered potential is precisely thesame as that of BECs for interacting atoms in the absence of an externalpotential.

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