In conclusion, we have found a rich structure in the spinwave modes of an
individual vortex, and have shown how these modes are related to the
instability of in-plane vortices to become out-of-plane vortices at
. This information can be valuable for improving the description
of the dynamics of interacting vortices in terms of their positions, and, the
internal vibrational motions we have found. Because this is a zero-temperature
single-vortex calculation, we can only speculate that it might be possible for
the vortex instability to affect dynamical correlation functions. If there is
an effect, it would be most prominent in the correlation function
of the out-of-plane spin components, especially for a
material whose anisotropy constant 
is near 
. It will
be a future challenge to consider whether the frequency spectra we have found
can be related to these dynamic correlations in thermal equilibrium in a
quantitative way.