Theory of degenerate Bose gas without anomalous averages
  Theory of a weakly non-ideal Bose gas in the canonical ensemble is developed without assumption of the C-number representation of the creation and annihilation operators with zero momentum. It is shown that the pole of the "density-density" Green's function exactly coincides with the Bogolybov's phonon-roton spectrum of excitations. At the same time, a gap exists in the one-particle excitation spectrum . This gap is related to the density of particles in the "condensate". 
V.B. Bobrov and S.A. Trigger
cond-mat.stat-mech cond-mat.quant-gas