  For a system of a large number of Bose particles, a chain of coupled
equations for the averages of field operators is obtained. In the approximation
where only the averages of one field operator and the averages of products of
two operators at zero temperature are taken into account, there is derived a
closed system of dynamic equations. Taking into account the finite range of the
interaction potential between particles, the spectrum of elementary excitations
of a many-particle Bose system is calculated, and it is shown that it has two
branches: a sound branch and an optical branch with an energy gap at zero
momentum. At high density, both branches are nonmonotonic and have the
roton-like minima. The dispersion of the phonon part of the spectrum is
considered. The performed calculations and analysis of experiments on neutron
scattering allow to make a statement about the complex structure of the Landau
dispersion curve in the superfluid He-4.
