where the coefficients are determined by and arbitrary parameters . The presence of non-scale coefficient is a special property of the eigenvalue problem (6); the ratio of amplitudes of in- and out-of-plane oscillations in the wave with given frequency is regulated by .
Far from the vortex core, , the coupling term in (6) is exponentially small, . For the solution for g can be expressed in the oscillating form (10). For one has asymptotics with linear combinations of exponents like , where , and constants depend on and . For only oscillating asymptotics like (10) appear.
In the most interesting region of small frequencies, , the eigenfunction with oscillations of combined with exponentially decaying can be constructed using the shooting method, ``killing'' the growing exponent in by choosing an appropriate value of . Numerically, the boundary condition was applied, and very well-pronounced exponential decay of resulted, even for . The scattering problem can be analyzed from these calculations. In order to explain let's consider the magnons without a vortex. The solutions can be represented in the form (7) with , , which is (10) with a defined value of the phase . In the presence of the vortex takes another value, which means the asympotic is in the form , where is a Neumann function. Obviously, the coefficient is the measure of the intensity of scattering, see Fig 4. Then, the S-matrix can be obtained from . The values of are smaller for larger values of m, for example, and for m=3,4 and 6.
It is well known that the presence of a quasi-local mode with large lifetime at gives a sharp maximum in , . The components of the S-matrix for all m's excluding have no such maxima at , and the maximum for is very wide. It means that there is no chance to have a well-defined quasi-local mode at for 1. However, at , the dependence is very fast, possibly indicating a root type singularity. The eigenfunctions near this point have a special shape with a well-defined maximum at , which is fitted by the translational mode functions ( ,) and a small amplitude oscillating tail. It means that the quasi-local mode with , is present, as in FMs.