Justification of the asymptotics of three one-dimensional short-range particles scattering problem solution for the processes $3\to2$

The present work is a continuation of the 3 body one-dimensional scattering problem in the presence of the bound states. Full description and justification of the generalized eigenfunction asymptotics in the case of the repulsive potentials have a simple geometric description. In the case of the presence of the bound states, additional terms appear in the eigenfunction asymptotics. In previous works during the analysis of Faddeev's equations in coordinate representation an operator of special form appeared which was connected to scattering amplitude and which did not have a simple description. In the current work, some properties of that operator are described. In particular, we derive the solvability of Faddeev's equations.