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We study magnon modes in the presence of a vortex in a circular easy-planeferromagnet. The problem of vortex-magnon scattering is investigated forpartial modes with different values of the azimuthal quantum number m over awide range of wave numbers. The analysis was done by combining analytical andnumerical calculations in the continuum limit with numerical diagonalization ofadequately large discrete systems. The general laws governing vortex-magnoninteractions are established. We give simple physical explanations of thescattering results: the splitting of doublets for the modes with opposite signsof $m$, which takes place for the long wavelength limit, is an analogue of theZeeman splitting in the effective magnetic field of the vortex. A singularbehavior for the scattering amplitude, $\sigma_m\propto k$, takes place as $k$diverges; it corresponds to the generalized Levinson theorem and can beexplained by the singular behavior of the effective magnetic field at theorigin.Comment: 14 pages (REVTeX), 4 figures (EPS

Amplitudes for magnon scattering by vortices in two-dimensional weakly easy-plane ferromagnets