Jost matrices for some analytically solvable potential models

Abstract

A family of analytically solvable potential models for the one- and two-channel problems is considered within the Jost matrix approach. The potentials are chosen to be constant in the interior region and to have different asymptotic behavior (tails) at large distances. The migration of the $S$-matrix poles on the Riemann surface of the energy, caused by variations of the potential strength, is studied. It is demonstrated that the long-range ($\ensuremath{\sim}1/{r}^{2}$) tails and Coulomb potential ($\ensuremath{\sim}1/r$) cause an unusual behavior of the $S$-matrix poles. It is found that in the two-channel problem with the long-range potentials the $S$-matrix poles may appear at complex energies on the physical Riemann sheet. The Coulomb tail not only changes the topology of the Riemann surface, but also breaks down the so-called mirror symmetry of the poles in both the single-channel and the two-channel problems.