  We calculate the magnon modes in the presence of a vortex in a circular
system, combining analytical calculations in the continuum limit with a
numerical diagonalization of the discrete system. The magnon modes are
expressed by the S-matrix for magnon-vortex scattering, as a function of the
parameters and the size of the system and for different boundary conditions.
Certain quasi-local translational modes are identified with the frequencies
which appear in the trajectory X(t) of the vortex center in recent Molecular
Dynamics simulations of the full many-spin model. Using these quasi-local modes
we calculate the two parameters of a 3rd-order quation of motion for X(t). This
equation was recently derived by a collective variable theory and describes
very well the trajectories observed in the simulations. Both parameters, the
vortex mass and a factor on the third time derivative of X(t), depend strongly
on the boundary conditions.
