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 ${\omega }_0$ and interacting with repulsive forces. The dispersion law $\omega ={\omega }_0({2n}^2+2n\ell +3n+\ell {)}^{1/2}$ for the elementary excitations is obtained for large numbers of atoms in the trap, to be compared with the prediction $\omega ={\omega }_0(2n+\ell )$ of the noninteracting harmonic oscillator model. Here $n$ is the number of radial nodes and $\ell$ 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.