Excited states of a dilute Bose-Einstein condensate in a harmonic trap

The low-lying hydrodynamic normal modes of a dilute Bose-Einstein gas in an isotropic harmonic trap determine the corresponding Bogoliubov amplitudes. In the Thomas-Fermi limit, these modes have large low-temperature occupation numbers and they permit an explicit construction of the dynamic structure function $S(\mathbf{q},\omega )$. The total noncondensate number $N^{\prime }\mathrm{}\left(0\right)\mathrm{}$ at zero temperature increases like $R^6$, where $R$ is the condensate radius measured in units of the oscillator length. The lowest dipole modes are constructed explicitly in the Bogoliubov approximation.