The Damping of the Bose-Condensate Oscillations in a Trap at Zero Temperature

We discuss an existence of the damping for the radial condensate oscillations in a cylindric trap at zero temperature. The damping is a result of the parametric resonance leading to energy transfer from the coherent condensate oscillations to the longitudinal sound waves within a finite frequency interval. The parametric resonance is due to the oscillations of the sound velocity. The triggering amplitudes at zero temperature are associated with the zero-point oscillations.