For m=0, the equations for f and g are uncoupled and have the forms of usual Schrödinger equations (SE). Then it is easy to show that the equation for has a gapless continuous spectrum only, with the usual oscillating asymptotic scattering form far from the vortex,
The equation for can be put in the form of a SE with the potential where at and at . So, the continuous spectrum of this problem has a gap, and for can be put in form (10) far from the vortex, with . But the presence of the attractive part of the potential, , gives the possibility of the appearance of a local mode with the frequency , , with well-pronounced exponential decay, . The analysis of this mode was done numerically by using a shooting method in finite circular systems of radius , with the conditions, , . Comparison with the results of exact diagonalization have shown very good agreement of the two approaches, see Fig. 3. The dependence of on R is very weak for large enough R, e.g., for , and for . Size effects become strong only for . The presence of a literally local mode inside the continuous spectrum is a unique property of AFM-vortices. It should manifest itself in response functions of the AFM's with vortices, and since the excitation of the local mode requires no momentum transfer, such resonance can in principle be observed in ESR experiments.