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For a system of a large number of Bose particles, a chain of coupledequations for the averages of field operators is obtained. In the approximationwhere only the averages of one field operator and the averages of products oftwo operators at zero temperature are taken into account, there is derived aclosed system of dynamic equations. Taking into account the finite range of theinteraction potential between particles, the spectrum of elementary excitationsof a many-particle Bose system is calculated, and it is shown that it has twobranches: a sound branch and an optical branch with an energy gap at zeromomentum. At high density, both branches are nonmonotonic and have theroton-like minima. The dispersion of the phonon part of the spectrum isconsidered. The performed calculations and analysis of experiments on neutronscattering allow to make a statement about the complex structure of the Landaudispersion curve in the superfluid He-4.

Phonon and optical-roton branches of excitations of the Bose system