Free energy of a many-boson system at low temperatures

The reduction of operators in the representation of collective variables to self-adjoint form is considered. The Hamiltonian and flux density operator of a many-boson system are reduced explicitly to self-adjoint form. For the obtained Hamiltonian, a perturbation theory is constructed in which each successive term contains, compared with the previous term, an extra sum over the wave vector. The free energy of a system of interacting Bose particles is calculated in the approximation of “two sums over the wave vectors”. From the free energy the internal energy is calculated, being represented as a quadratic functional of the mean population numbers of the elementary excitations. At the same time, the temperature-dependent correction to the Bogolyubov energy spectrum of the elementary excitations is obtained.