  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.
