Collective excitations and the gap in spectrum of the degenerated Bose gas

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Model of the degenerated weakly non-ideal Bose gas is considered without suggestion on C-number representation of the creation and annihilation operators with zero momentum. The "density-density" correlation function and the one-particle Green function are calculated on basis of the suggestion about C-number representation of the operator of particle density in the Bose condensate. It is shown, that the pole in the "density-density" Green function determines the Bogolyubov's spectrum of the collective excitations, which have the phonon-roton form. At the same time the spectrum of the one-particle excitations possesses a gap, whose value is connected with the density of particles in the Bose condensate.