Exactness of the Bogoliubov approximation in random external potentials

Abstract

We investigate the validity of the Bogoliubov c-number approximation in the case of interacting Bose-gas in a \textit{homogeneous random} media. To take into account the possible occurence of type III generalized Bose-Einstein condensation (i.e. the occurrence of condensation in an infinitesimal band of low kinetic energy modes without macroscopic occupation of any of them) we generalize the c-number substitution procedure to this band of modes with low momentum. We show that, as in the case of the one-mode condensation for translation-invariant interacting systems, this procedure has no effect on the exact value of the pressure in the thermodynamic limit, assuming that the c-numbers are chosen according to a suitable variational principle. We then discuss the relation between these c-numbers and the (total) density of the condensate

On the nature of Bose–Einstein condensation enhanced by localization

“…Hence, the (generalized) condensation produced in these models by the localization property of the one-particle Schrödinger operator can be correctly described as of "Bose-Einstein" type in the traditional sense. This opens up the possibility of formulating a generalized version of the c-number Bogoliubov approximation ( [9], [10]). In the case of the weak external potential, perhaps this result is not so surprising since the model is asymptotically translation invariant, but in the random case, it is less obvious since the system is translation invariant only in the sense that translates of the potential are equally probable and therefore for a given configuration the system is not translation invariant.…”