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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n right-arrow normal infinity"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo stretchy="false"> → </mml:mo> <mml:mspace width="thinmathspace"/> <mml:mi mathvariant="normal"> ∞ </mml:mi> </mml:mrow> <mml:annotation encoding="application/x-tex">n\, \to \,\infty</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , to the solution of one of the recurrence formulas with the boundary conditions given at <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="n equals 0"> <mml:semantics> <mml:mrow> <mml:mi>n</mml:mi> <mml:mspace width="thinmathspace"/> <mml:mo>=</mml:mo> <mml:mspace width="thinmathspace"/> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">n\, = \,0</mml:annotation> </mml:semantics> </mml:math> </inline-formula> . 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.