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].

Wed Sep 6 18:51:57 CDT 1995