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 ,
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 ,.