Phonon-Phonon Interactions and Propagation of Sound in Liquid He-4

The quantum-hydrodynamical model for excitations in He II at low temperatures is used to calculate the velocity shift and the attenuation of ultrasound in the collisionless domain. First, starting from a linear spectrum, the self-energy needed for the evaluation of the density-density correlation function is derived in lowest order of perturbation theory. Particular attention is then paid to a self-consistent determination of the real part of the pole of the retarded Green's function. This procedure turns out to be essential for this model. Second, the influence of a nonlinear spectrum and of a finite lifetime of thermal phonons is investigated.