Analytic properties of an exactly solvable nonrelativistic two-channel scattering problem

A study is made of an exactly solvable example of a nonrelativistic two-channel scattering problem of two neutral spinless particles. The $S$-matrix of the problem is found and the following result is obtained. If the $S$-matrix is represented as the ratio of Jost matrices, the assumption $S_{12}=S_{21}=0$ and $S_{22}=1$ made in [4] [Teor. Mat. Fiz., 4, 270 (1970)] concerning the behavior of the elements of the $S$-matrix need not be satisfied in the region between the thresholds. A connection between the fulfillment of these conditions and $T$ invariance of the Hamiltonian is revealed. It is shown that the wave function can have singularities on the real energy axis in the unphysical region between the thresholds. The example of a simple pole is taken to demonstrate the influence of such a singularity on the behavior of all the cross sections in the region of the threshold of the upper channel.