This chapter examines the properties of one-dimensional Jost solutions for S-matrix problems. It first considers how the left–right transmission and reflections coefficients can be expressed in terms of the elements of the S-matrix for one-dimensional scattering problems on, focusing on poles of the transmission coefficient. It then uses the radial equation to revisit the problem of the square-well potential from the perspective of the Jost solution, with Jost boundary conditions at r = 0 and as r approaches infinity. It also presents the notations for the Jost functions and the S-matrix before discussing the problem of scattering from a constant spherical inhomogeneity.
An asimptotic expansion for the Green’s function of nonrelativistic potential-scattering theory and the asymptotic character of the born series for the Jost function for large complex angular momentum