Orthogonal polynomials from the viewpoint of scattering theory

Abstract

It is demonstrated that there is a close parallel between the theory of a class of orthogonal polynomials and scattering theory. In both cases a fundamental role is played by a particular solution of the basic difference (differential) equation which we call the Jost function. Under rather general conditions this function has simple analytic properties. It determines and is largely determined by either the asymptotic phases or the continuous part of the weight (spectral) function. Indeed this is more than an analogy. By appropriate limiting procedures one can pass from a result about orthogonal polynomials to one in scattering theory. Conversely, scattering theory throws considerable light on theorems about orthogonal polynomials.