Fuda's off-shell Jost function for Coulomb, Hulthén, and Eckart potentials and limiting relations

We study the off-shell Jost function $f(k, q)$, introduced by Fuda, for the Coulomb, the Hulthén, and two modified Eckart potentials. A simple closed expression for the $l=0\mathrm{}$ Coulomb off-shell Jost function has been obtained. This function is discontinuous at $q=k$. Its on-shell limiting behavior is given by the singular factor ${\mathrm{}(q-k)\mathrm{}}^{-i\gamma }$, where $\gamma$ is the Sommerfeld parameter. We also discuss the off-shell Jost solution $f(k, q, r)$, which is an off-shell generalization of the Jost solution $f(k, r)$. We consider the Hulthén potential as a screened Coulomb potential, let the screening parameter $a$ go to infinity, and derive the limiting behavior of the Jost solution, the Jost function, the off-shell Jost function, and the half-shell $T$ matrix for the Hulthén potential as $a\to \infty$. We obtain discontinuities given by the singular factor $a^{i\gamma }$. For comparison, we introduce two modifications of an Eckart potential which can be considered to be a screened $r^{-2\mathrm{}}$ potential and derive a number of limiting relations in analogy to those for the Hulthén-Coulomb pair of potentials.