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.

Wed Sep 6 18:51:57 CDT 1995