Temperature dependent shifts in collective excitations of a gaseous Bose-Einstein condensate

The moment method is applied to the quantum theory for a trapped dilute gas, obtaining equations for the evolution of the cloud. These equations proof the existence of undamped oscillations in a two-dimensional harmonic trap with radial symmetry. In all other cases we can, with physically reasonable approximations, solve the equations and find the frequencies of the collective modes for all temperatures, from the pure Bose-Einstein condensate up to the normal gas.