Spectral and dispersion properties of pair soliton states in a quasi-one-dimensional anisotropic antiferromagnet with $S=1/2$

By solution of the Schrödinger equation in the continuum approximation, it is shown analytically that there exist excited eigenstates of the quasi-one-dimensional Ising antiferromagnet with spin $S=1/2$ in the form of spatially localized quantum states. Computer modeling of a discrete model of interacting solitons with allowance for the symmetry of the solutions gives eigenvalues of the Sturm sequence that differ from the solutions of the continuum approximation. The spectral and dispersion properties of the nonlinear bound states of lowest energy and the selection rules in resonance transitions in an external magnetic field applied parallel to and perpendicular to the axis of magnetic anisotropy are calculated.