We calculate the magnon modes in the presence of a vortex in a circularsystem, combining analytical calculations in the continuum limit with anumerical diagonalization of the discrete system. The magnon modes areexpressed by the S-matrix for magnon-vortex scattering, as a function of theparameters and the size of the system and for different boundary conditions.Certain quasi-local translational modes are identified with the frequencieswhich appear in the trajectory X(t) of the vortex center in recent MolecularDynamics simulations of the full many-spin model. Using these quasi-local modeswe calculate the two parameters of a 3rd-order quation of motion for X(t). Thisequation was recently derived by a collective variable theory and describesvery well the trajectories observed in the simulations. Both parameters, thevortex mass and a factor on the third time derivative of X(t), depend stronglyon the boundary conditions.
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