  The Bogoliubov approximation is used to study the excited states of a dilute
gas of $N$ atomic bosons trapped in an isotropic harmonic potential
characterized by a frequency $\omega_0$ and an oscillator length $d_0 =
\sqrt{\hbar/m\omega_0}$. The self-consistent static Bose condensate has
macroscopic occupation number $N_0 \gg 1$, with nonuniform spherical condensate
density $n_0(r)$; by assumption, the depletion of the condensate is small ($N'
\equiv N - N_0\ll N_0$). The linearized density fluctuation operator $\hat
\rho'$ and velocity potential operator $\hat\Phi'$ satisfy coupled equations
that embody particle conservation and Bernoulli's theorem. For each angular
momentum $l$, introduction of quasiparticle operators yields coupled eigenvalue
equations for the excited states; they can be expressed either in terms of
Bogoliubov coherence amplitudes $u_l(r)$ and $v_l(r)$ that determine the
appropriate linear combinations of particle operators, or in terms of
hydrodynamic amplitudes $\rho_l'(r)$ and $\Phi_l'(r)$. The hydrodynamic picture
suggests a simple variational approximation for $l >0$ that provides an upper
bound for the lowest eigenvalue $\omega_l$ and an estimate for the
corresponding zero-temperature occupation number $N_l'$; both expressions
closely resemble those for a uniform bulk Bose condensate.
