Collective Excitations of a Trapped Bose-Condensed Gas

We investigate the low energy excitations of a dilute atomic Bose gas confined in a harmonic trap of frequency ${\ensuremath{\omega}}_{0}$ and interacting with repulsive forces. The dispersion law $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}({2n}^{2}+2n\ensuremath{\ell}+3n+\ensuremath{\ell}{)}^{1/2}$ for the elementary excitations is obtained for large numbers of atoms in the trap, to be compared with the prediction $\ensuremath{\omega}\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\omega}}_{0}(2n+\ensuremath{\ell})$ of the noninteracting harmonic oscillator model. Here $n$ is the number of radial nodes and $\ensuremath{\ell}$ is the orbital angular momentum. The effects of the kinetic energy pressure are estimated using a sum rule approach. Results are also presented for deformed traps and attractive forces.