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[15] 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.
