Microscopic theory of the energy spectrum of liquid HeII

The wave function $\psi_k$ of the lowest excites state of many-boson system is represented as the product of the wave function of the ground state $\psi_0=e^U$ and the function $\varphi_k$ which takes into account the existence of the excitation with the momentum $\hbar k$ and the energy $E(k)$. The function $U$ and $\varphi_k$ are represented as the series over the collective variables which are the Fourier coefficients of the operator of density fluctuations. The systems of coupled equations for the coefficient functions of these series are derived. The system of two integral equations connecting the excitation spectrum $E(k)$ with the structure factor of the system is obtained by means of breaking the chain of equations. This makes it possible to apply the results obtained to the liquid $\mathrm{He}^4$. The damping of the spectrum is described. It is shown that the expansion of the spectrum in the limit of small momentum $\hbar k$ includes the odd powers of $\hbar k$ only.