On the integral representations of the Jost function and Coulomb off-shell Jost solution

Abstract

The integral representations of the Jost function (on-and off-shell) are rederived by the judicious use of the transposed operator relation on the particular integrals for Jost solution and using one of these particular integrals an analytical expression for the Coulomb off-shell Jost solution is presented in the maximal reduced form.It is well-known that in charged-particle scattering the long range of Coulomb interaction is a source of special difficulties due to 1/r behaviour of the potential at large distances. These difficulties are observed both in classical and quantum scattering. The scattering wave functions for charged-particle scattering exhibit an essentially different (logarithmic) behaviour at large distances compared to scattering wave functions for short-range potentials. This leads to a discontinuity in the behaviour of half-and off-shell functions for the Coulomb and Coulomb-like interactions at the energy shell. The characteristic Coulomb discontinuity arises from the fact that a long-range potential distorts not only the scattering wave but also the incident plane wave [1]. However, in such a situation, one can extract physical information by introducing suitable Coulombian asymptotic states [2].The behaviour of the irregular solution f (k, r) of the radial Schrödinger equation near the origin determines the Jost function [3] f (k) which has an important role in examining the analytic properties of partial wave scattering amplitude. The Jost function has two integral representations [4]: one in terms of irregular solution f (k, r) and the other in terms of the regular solution φ (k, r). The integral representation associated with irregular solution f (k, r) follows directly from the integral equation for f (k, r). Contrary to this, the other integral representation is derived with particular emphasis on the asymptotic behaviour of regular solution φ (k, r).The off-shell Jost function [5] f (k, q), with q, an off-shell momentum, is also determined from the irregular solution f (k, q, r) of an inhomogeneous Schrödinger 357

Regular and Jost states for the S-wave Manning–Rosen potential

“…, of the radial Schrödinger equation near the origin determines the Jost function ( ) I + k , which has an important role in examining the analytical properties of the partial wave scattering amplitude [21,[32][33][34][35]. In the upper half of the complex k-plane, > k Im 0, ( )…”