Inverse problem of the potential scattering of charged particles at fixed orbital angular momentum and energy

Expressions are obtained for recovering the central interaction potential of charged particles in the case when the interaction leads to multiplication of the $S$ matrix defined in the complex plane of the Coulomb coupling constant by a rational $S$ function. A study is made of the influence of the sign of the potential on the position of the poles of the $S$ function and of the behavior of the phase shift.