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

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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 , to the solution of one of the recurrence formulas with the boundary conditions given at . 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.

**[1]**Z. S. Agranovich and V. A. Marchenko,*The inverse problem of scattering theory*, Translated from the Russian by B. D. Seckler, Gordon and Breach Science Publishers, New York-London, 1963. MR**0162497****[2]**N. I. Akhiezer,*The classical moment problem and some related questions in analysis*, Translated by N. Kemmer, Hafner Publishing Co., New York, 1965. MR**0184042****[3]**Glen Baxter,*A convergence equivalence related to polynomials orthogonal on the unit circle*, Trans. Amer. Math. Soc.**99**(1961), 471–487. MR**0126126**, https://doi.org/10.1090/S0002-9947-1961-0126126-8**[4]**K. M. Case,*Orthogonal polynomials from the viewpoint of scattering theory*, J. Mathematical Phys.**15**(1974), 2166–2174. MR**0353860**, https://doi.org/10.1063/1.1666597**[5]**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**0332067**, https://doi.org/10.1063/1.1666237**[6]**J. S. Geronimo,*Scattering theory and orthogonal polynomials*, Doctoral Dissertation, Rockefeller Univ., 1977.**[7]**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**, https://doi.org/10.1063/1.524077**[8]**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****[9]**Paul G. Nevai,*Orthogonal polynomials*, Mem. Amer. Math. Soc.**18**(1979), no. 213, v+185. MR**519926**, https://doi.org/10.1090/memo/0213**[10]**G. Szegö,*Orthogonal polynomials*, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1939, p. 27; 4th edition, 1975.**[11]**A. Zygmund,*Trigonometric series. 2nd ed. Vols. I, II*, Cambridge University Press, New York, 1959. MR**0107776**

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

Article copyright:
© Copyright 1980
American Mathematical Society