Validity of the Hohenberg theorem for a generalized Bose-Einstein condensation in two dimensions

Several authors have considered the possibility of a generalized Bose-Einst ein condensation (BEC) in which a band of low states is occupied so that th e total occupation number is macroscopic, even if the occupation number of each state is not extensive. The Hohenberg theorem (HT) states that there i s no BEC into a single state in 2D; we consider its validity for the case o f a generalized condensation and find that, under certain conditions, the H T does not forbid a BEC in 2D. We discuss whether this situation actually o ccurs in any theoretical model system.