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 ω₀ and interacting with repulsive forces. The dispersion law ω = ω₀(2n²+2nℓ+3n+ℓ)^1/2 for the elementary excitations is obtained for large numbers of atoms in the trap, to be compared with the prediction ω = ω₀(2n+ℓ) of the noninteracting harmonic oscillator model. Here n is the number of radial nodes and ℓ 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.