  We study magnon modes in the presence of a vortex in a circular easy-plane
ferromagnet. The problem of vortex-magnon scattering is investigated for
partial modes with different values of the azimuthal quantum number m over a
wide range of wave numbers. The analysis was done by combining analytical and
numerical calculations in the continuum limit with numerical diagonalization of
adequately large discrete systems. The general laws governing vortex-magnon
interactions are established. We give simple physical explanations of the
scattering results: the splitting of doublets for the modes with opposite signs
of $m$, which takes place for the long wavelength limit, is an analogue of the
Zeeman splitting in the effective magnetic field of the vortex. A singular
behavior for the scattering amplitude, $\sigma_m\propto k$, takes place as $k$
diverges; it corresponds to the generalized Levinson theorem and can be
explained by the singular behavior of the effective magnetic field at the
origin.
