Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 085, 16 pages      arXiv:1706.02391      https://doi.org/10.3842/SIGMA.2017.085
Contribution to the Special Issue on Orthogonal Polynomials, Special Functions and Applications (OPSFA14)

The Inverse Spectral Problem for Jacobi-Type Pencils

Sergey M. Zagorodnyuk
School of Mathematics and Computer Sciences, V.N. Karazin Kharkiv National University, Svobody Square 4, Kharkiv 61022, Ukraine

Received June 10, 2017, in final form October 24, 2017; Published online October 28, 2017

Abstract
In this paper we study the inverse spectral problem for Jacobi-type pencils. By a Jacobi-type pencil we mean the following pencil $J_5 - \lambda J_3$, where $J_3$ is a Jacobi matrix and $J_5$ is a semi-infinite real symmetric five-diagonal matrix with positive numbers on the second subdiagonal. In the case of a special perturbation of orthogonal polynomials on a finite interval the corresponding spectral function takes an explicit form.

Key words: operator pencil; recurrence relation; orthogonal polynomials; spectral function.

pdf (361 kb)   tex (19 kb)

References

  1. Atkinson F.V., Discrete and continuous boundary problems, Mathematics in Science and Engineering, Vol. 8, Academic Press, New York - London, 1964.
  2. Ben Amara J., Shkalikov A.A., Vladimirov A.A., Spectral and oscillatory properties of a linear pencil of fourth-order differential operators, Math. Notes 94 (2013), 49-59, arXiv:1112.2351.
  3. Berezans'kiǐ Ju.M., Expansions in eigenfunctions of selfadjoint operators, Translations of Mathematical Monographs, Vol. 17, Amer. Math. Soc., Providence, RI, 1968.
  4. Chihara T.S., An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York - London - Paris, 1978.
  5. Gherbi A., Messirdi B., Benharrat M., Quotient operators: new generation of linear operators, Funct. Anal. Approx. Comput. 7 (2015), 85-93.
  6. Ismail M.E.H., Classical and quantum orthogonal polynomials in one variable, Encyclopedia of Mathematics and its Applications, Vol. 98, Cambridge University Press, Cambridge, 2005.
  7. Markus A.S., Introduction to the spectral theory of polynomial operator pencils, Translations of Mathematical Monographs, Vol. 71, Amer. Math. Soc., Providence, RI, 1988.
  8. Möller M., Pivovarchik V., Spectral theory of operator pencils, Hermite-Biehler functions, and their applications, Operator Theory: Advances and Applications, Vol. 246, Birkhäuser/Springer, Cham, 2015.
  9. Parlett B.N., The symmetric eigenvalue problem, Classics in Applied Mathematics, Vol. 20, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1998.
  10. Rodman L., An introduction to operator polynomials, Operator Theory: Advances and Applications, Vol. 38, Birkhäuser Verlag, Basel, 1989.
  11. Simon B., Szegő's theorem and its descendants. Spectral theory for $L^2$ perturbations of orthogonal polynomials, M.B. Porter Lectures, Princeton University Press, Princeton, NJ, 2011.
  12. Suetin P.K., Classical orthogonal polynomials, 3rd ed., Fizmatlit, Moscow, 2005.
  13. Szegő G., Orthogonal polynomials, American Mathematical Society, Colloquium Publications, Vol. 33, 4th ed., Amer. Math. Soc., Providence, RI, 1975.
  14. Teschl G., Jacobi operators and completely integrable nonlinear lattices, Mathematical Surveys and Monographs, Vol. 72, Amer. Math. Soc., Providence, RI, 2000.
  15. Zagorodnyuk S.M., The direct and inverse spectral problems for some banded matrices, Serdica Math. J. 37 (2011), 9-24.
  16. Zagorodnyuk S.M., Orthogonal polynomials associated with some Jacobi-type pencils, Ukrain. Math. J. 68 (2016), 1353-1365, arXiv:1508.01794.

Previous article  Next article   Contents of Volume 13 (2017)