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.