Coexistence of “bogolons” and the single-particle excitation spectrum with a gap in the degenerate Bose gas

Properties of the weakly non-ideal Bose gas are considered without suggestion on the C-number representation of the creation and annihilation operators with zero momentum. The “density–density” correlation function and the one-particle Green's function of the degenerate Bose gas are calculated based on the self-consistent Hartree–Fock approximation. It is shown that the single-particle excitation spectrum has a gap associated with the particle density in the “condensate”. At the same time, the pole in the “density–density” Green function determines the phonon–roton excitation spectrum which is identical to that discovered by Bogolyubov for collective excitations (“bogolons”).