Vibrations of a columnar vortex in a trapped Bose-Einstein condensate

We derive a governing equation for a Kelvin wave supported on a vortex line in a Bose-Einstein condensate, in a rotating cylindrically symmetric parabolic trap, where it is assumed that the shape of the vortex line is dominated by the properties of the condensate at the center of the trap. From this solution the Kelvin wave dispersion relation is determined. In the limit of an oblate trap and in the absence of longitudinal trapping our results are consistent with previous work. We show that the derived Kelvin wave dispersion in the general case is in quantitative agreement with numerical calculations of the Bogoliubov spectrum.