  It is demonstrated theoretically that the counter-rotating quadrupole mode in
a vortex of Bose-Einstein condensates can decay into a pair of Kelvin modes via
Beliaev process. We calculate the spectral weight of density-response function
within Bogoliubov framework, taking account of both Beliaev and Landau
processes. Good agreement with experiment on $^{87}$Rb by Bretin et al.
[cond-mat/0211101] allows us to unambigiously identify the decayed mode as the
Kelvin wave propagating along a vortex line.
