Scattering theory and polynomials orthogonal on the real line

Geronimo, J. S.; Case, K. M.

doi:10.1090/S0002-9947-1980-0558185-4

Scattering theory and polynomials orthogonal on the real line
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by J. S. Geronimo and K. M. Case

Trans. Amer. Math. Soc. 258 (1980), 467-494

DOI: https://doi.org/10.1090/S0002-9947-1980-0558185-4

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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 $n \to \infty$, to the solution of one of the recurrence formulas with the boundary conditions given at $n = 0$. 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.

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