Abstract
We calculate the magnon modes in the presence of a vortex on 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 quasilocal translational modes are identified with the frequencies which appear in the trajectory $\stackrel{\ensuremath{\rightarrow}}{X}(t)$ of the vortex center in recent molecular dynamics simulations of the full many-spin model. Using these quasilocal modes we calculate the two parameters of a third-order equation of motion for $\stackrel{\ensuremath{\rightarrow}}{X}(t).$ 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 the factor in front of $\stackrel{\ensuremath{\rightarrow}}{X}⃛,$ depend strongly on the boundary conditions.
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