General properties of potentials for which the Schrödinger equation can be solved by means of hypergeometric functions

A general investigation is made into the problem of constructing (for zero angular momentum) the Jost function and $S$ matrix for a family of, potentials that admit solution of the radial Schrödinger equation by means of hypergeometric series. The passage to the limit of the confluent hypergeometric equation leads to potentials with Coulomb asymptotic behavior at infinity. For these potentials, a general expression for the Green's function is given in the form of the product of two Whittaker functions. This makes it possible to gather together the results obtained by a number of authors for potentials of special form specified as explicit functions of the variable $r$.