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# Magnons on a vortex. Continuous approach.

Let's introduce small deviations of variables from those in the vortex solution, In local rotated coordinate frames, , where the axis coincides with the -vector's direction in the unperturbed vortex, and

the variables are the projections of the -vector on the local axes and : .

It is easy to get the coupled set of two partial differential equations for and :

where , and the ``potentials'' are:\ . Using the ansatz like that in [8],

where and a,b are arbitrary constants, one obtains a coupled set of ODEs for and . Here ...; k and m are the full set of quantum numbers denoting the eigenvalues The presence of the combination of exponentials with arbitrary coefficients is due to the degeneracy of the modes with m=+|m| and m=-|m|. One can see that the coupling of and in Eq. (6) comes from the term with only. This means that (i) the coupling vanishes exponentially at (); (ii) there is no coupling for m=0 modes, in contrast to the FM case [8],[12].

Gary M Wysin
Wed Sep 6 18:51:57 CDT 1995