Inverse scattering—exact solution of the Gel’fand-Levitan equation

One formulation of inverse scattering theory involves the Gel’fand–Levitan equation. We present a procedure for finding exact solutions to this equation; this procedure can be applied whenever the reflection coefficient is a rational function of the wave number k, with an arbitrary number of poles. We present graphs of the potential as a function of distance, for several cases with 3, 4, 5, and 6 poles. Prior to this paper, no 4-, 5-, or 6-pole case had ever been treated successfully.