Scattering theory and polynomials orthogonal on the real line

Authors:
J. S. Geronimo and K. M. Case

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

MSC:
Primary 81F99; Secondary 30C10, 42C05

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

MathSciNet review:
558185

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

- Z. S. Agranovich and V. A. Marchenko,
*The inverse problem of scattering theory*, Gordon and Breach Science Publishers, New York-London, 1963. Translated from the Russian by B. D. Seckler. MR**0162497** - N. I. Akhiezer,
*The classical moment problem and some related questions in analysis*, Hafner Publishing Co., New York, 1965. Translated by N. Kemmer. MR**0184042** - Glen Baxter,
*A convergence equivalence related to polynomials orthogonal on the unit circle*, Trans. Amer. Math. Soc.**99**(1961), 471–487. MR**126126**, DOI https://doi.org/10.1090/S0002-9947-1961-0126126-8 - K. M. Case,
*Orthogonal polynomials from the viewpoint of scattering theory*, J. Mathematical Phys.**15**(1974), 2166–2174. MR**353860**, DOI https://doi.org/10.1063/1.1666597 - K. M. Case and S. C. Chiu,
*The discrete version of the Marchenko equations in the inverse scattering problem*, J. Mathematical Phys.**14**(1973), 1643–1647. MR**332067**, DOI https://doi.org/10.1063/1.1666237
J. S. Geronimo, - J. S. Geronimo and K. M. Case,
*Scattering theory and polynomials orthogonal on the unit circle*, J. Math. Phys.**20**(1979), no. 2, 299–310. MR**519213**, DOI https://doi.org/10.1063/1.524077 - G. Š. Guseĭnov,
*Determination of an infinite Jacobi matrix from scattering data*, Dokl. Akad. Nauk SSSR**227**(1976), no. 6, 1289–1292 (Russian). MR**0405160** - Paul G. Nevai,
*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, DOI https://doi.org/10.1090/memo/0213
G. Szegö, - A. Zygmund,
*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

*Scattering theory and orthogonal polynomials*, Doctoral Dissertation, Rockefeller Univ., 1977.

*Orthogonal polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939, p. 27; 4th edition, 1975.

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
81F99,
30C10,
42C05

Retrieve articles in all journals with MSC: 81F99, 30C10, 42C05

Additional Information

Article copyright:
© Copyright 1980
American Mathematical Society