Quantized Vortices and Collective Oscillations of a Trapped Bose-Einstein Condensate

Using a sum rule approach we calculate the frequency shifts of the quadrupole oscillations of a harmonically trapped Bose gas due to the presence of a quantized vortex. Analytic results are obtained for positive scattering lengths and large $N$ where the shift relative to excitations of opposite angular momentum is found to be proportional to the quantum circulation of the vortex and to decrease as $N^{-2\mathrm{}/5}$. Results are also given for smaller values of $N$ covering the transition between the ideal gas and the Thomas-Fermi limit. The splitting of the collective frequencies in toroidal configurations is also discussed.