A three-magnon problem for exactly rung-dimerized spin ladders: from a general outlook to the Bethe ansatz

A three-magnon problem for an exactly rung-dimerized spin ladder is brought up separately in all total spin sectors. At first, a special duality transformation of the Schrodinger equation is found within the general outlook. Then the problem is treated within the coordinate Bethe ansatz. A straightforward approach is developed to obtain pure scattering states. At values S = 0 and S = 3 of total spin, the Schrodinger equation has a form inherent in the XXZ chain. At S = 1, 2, solvability holds only in five previously found completely integrable cases. Nevertheless, even in a general non-integrable case, there are some special Bethe solutions in both S = 1 and S = 2 sectors. Pure scattering states in all total spin sectors are presented explicitly.