Branch structure of the Bose-condensate excitations spectrum

The two-branch excitation spectrum of the non-ideal Bose-gas is obtained by the Path Integral method for the canonical distribution with constraint. Reorganization of the spectrum happens for a value of the condensate density, providing the balance between attraction and repulsion forces of the interaction potential. Two branches show different dependence on the temperature, Bose-condensate density and stability relative to the quantum fluctuations of atom amplitudes. A qualitative similarity of the properties of these branches with the recently observed experimental structure of neutron sharp peak scattering is found