Two-magnon states in a one-dimensional Heisenberg model with second-neighbor interaction

The two-magnon problem is solved for a chain of spins $s=1/2$ with allowance for the interaction of neighbors that come after the nearest neighbors. It is shown that for any interaetion of the second neighbors (all the interactions are ferromagnetic) a bound state of two spin waves exists for all $k$, $0<k<\pi$ ($k$ is the center of mass momentum of the system of two magnons). For $k=\pi$ a sufficiently strong interaction of the second neighbors destroys the bound state, and this result eonfirms the results of [3, 4].