We study the perfect Bose gas in random external potentials and show that there is generalized Bose-Einstein condensation in the random eigenstates if and only if the same occurs in the one-particle kinetic-energy eigenstates, which corresponds to the generalized condensation of the free Bose gas. Moreover, we prove that the amounts of both condensate densities are equal. Our method is based on the derivation of an explicit formula for the occupation measure in the one-body kinetic-energy eigenstates which describes the repartition of particles among these non-random states. This technique can be adapted to re-examine the properties of the perfect Bose gas in the presence of weak (scaled) non-random potentials, for which we establish similar results.
In a previous paper we established that for the perfect Bose gas and the mean-field Bose gas with an external random or weak potential, whenever there is generalized Bose-Einstein condensation in the eigenstates of the single particle Hamiltonian, there is also generalized condensation in the kinetic energy states. In these cases Bose-Einstein condensation is produced or enhanced by the external potential. In the present paper we establish a criterion for the absence of condensation in single kinetic energy states and prove that this criterion is satisfied for a class of random potentials and weak potentials. This means that the condensate is spread over an infinite number of states with low kinetic energy without any of them being macroscopically occupied.
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
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