  We study the effects of repulsive interaction and disorder on Bosons in a
two-site Bose-Hubbard system, which provides a simple model of the dirty boson
problem. By comparison with exact numerical results, we demonstrate how a
straightforward application of the Bogoliubov approximation fails even to
deliver a qualitatively correct picture: It wrongly predicts an increase of the
condensate depletion due to disorder. We show that, in the presence of
disorder, the noncommutative character of the condensate operator has to be
retained for a correct description of the system.
