Collective excitations of weakly coupled Bose condensates confined in harmonic traps

We use ordinary first-order perturbation theory in the mean-field interaction to derive analytic expressions for the frequencies of the excitation modes of a dilute Bose-Einstein condensate under confinement. The confining trap is modelled by a harmonic potential having either spherical or cylindrical symmetry. The expressions obtained are valid for all quantum numbers of the appropriate symmetry and for attractive as well as repulsive effective interactions. Comparison with available numerical calculations is made.