Scattering theory and polynomials orthogonal on the real line

Abstract

The techniques of scattering theory are used to study polynomials orthogonal on a segment of the real line. Instead of applying these techniques to the usual three-term recurrence formula, we derive a set of two two-term recurrence formulas satisfied by these polynomials. One of the advantages of these new recurrence formulas is that the Jost function is related, in the limit as n → ∞ n\, \to \,\infty , to the solution of one of the recurrence formulas with the boundary conditions given at n = 0 n\, = \,0 . In this paper we investigate the properties of the Jost function and the spectral function assuming the coefficients in the recurrence formulas converge at a particular rate.