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

Abstract

We study the off-shell Jost function $f(k, q)$, introduced by Fuda, for the Coulomb, the Hulth\'en, and two modified Eckart potentials. A simple closed expression for the $l=0$ 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 ${(q\ensuremath{-}k)}^{\ensuremath{-}i\ensuremath{\gamma}}$, where $\ensuremath{\gamma}$ 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 Hulth\'en 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 Hulth\'en potential as $a\ensuremath{\rightarrow}\ensuremath{\infty}$. We obtain discontinuities given by the singular factor ${a}^{i\ensuremath{\gamma}}$. For comparison, we introduce two modifications of an Eckart potential which can be considered to be a screened ${r}^{\ensuremath{-}2}$ potential and derive a number of limiting relations in analogy to those for the Hulth\'en-Coulomb pair of potentials.