Kelvin mode of a vortex in a nonuniform Bose-Einstein condensate

In a uniform fluid, a quantized vortex line with circulation h/M can support long-wavelength helical traveling waves proportional to e^{i(kz-\omega_k t)} with the well-known Kelvin dispersion relation \omega_k \approx (\hbar k^2/2M) \ln(1/|k|\xi), where \xi is the vortex-core radius. This result is extended to include the effect of a nonuniform harmonic trap potential, using a quantum generalization of the Biot-Savart law that determines the local velocity V of each element of the vortex line. The normal-mode eigenfunctions form an orthogonal Sturm-Liouville set. Although the line's curvature dominates the dynamics, the transverse and axial trapping potential also affect the normal modes of a straight vortex on the symmetry axis of an axisymmetric Thomas-Fermi condensate. The leading effect of the nonuniform condensate density is to increase the amplitude along the axis away from the trap center. Near the ends, however, a boundary layer forms to satisfy the natural Sturm-Liouville boundary conditions. For a given applied frequency, the next-order correction renormalizes the local wavenumber k(z) upward near the trap center, and k(z) then increases still more toward the ends.