Collective excitations of a trapped Bose-Einstein condensate at finite large atom numbers

We study the collective excitations of a trapped Bose-Einstein condensate for a large but finite number of atoms. Since our calculations go beyond the Thomas-Fermi approximation, the "quantum pressure" and the change of chemical potential due to the density fluctuations must be considered. The small kinetic energy enhancement and chemical potential reduction can't be neglected with respect to the strong inter-atomic interaction. The hydrodynamic equation is corrected for the large but finite number of atoms and its solution is obtained from perturbation theory. A variational calculation of the ground-state density yields explicit analytic expressions for all excitation energies as functions of the interaction strength for harmonic traps with spherical and cylindrical symmetries. Within the wide interaction ranges where the condition of perturbation calculation is satisfied, the calculated results agree well with those of the experiments and simulations.