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Special Matrices

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The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix

F. Štampach
  • Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague, Kolejní 2, 16000 Praha, Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ P. Šťovíček
  • Department of Mathematics, Faculty of Nuclear Science, Czech Technical University in Prague, Trojanova 13, 12000 Praha, Czech Republic
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
Published Online: 2014-10-01 | DOI: https://doi.org/10.2478/spma-2014-0014

Abstract

A family T(ν), ν ∈ ℝ, of semiinfinite positive Jacobi matrices is introduced with matrix entries taken from the Hahn-Exton q-difference equation. The corresponding matrix operators defined on the linear hull of the canonical basis in ℓ2(ℤ+) are essentially self-adjoint for |ν| ≥ 1 and have deficiency indices (1, 1) for |ν| < 1. A convenient description of all self-adjoint extensions is obtained and the spectral problem is analyzed in detail. The spectrum is discrete and the characteristic equation on eigenvalues is derived explicitly in all cases. Particularly, the Hahn-Exton q-Bessel function Jν(z; q) serves as the characteristic function of the Friedrichs extension. As a direct application one can reproduce, in an alternative way, some basic results about the q-Bessel function due to Koelink and Swarttouw.

Keywords: Jacobi matrix; Hahn-Exton q-Bessel function; self-adjoint extension; spectral problem

References

  • [1] L. D. Abreu, J. Bustoz, J. L. Cardoso: The roots of the third Jackson q-Bessel function, Internat. J. Math. Math. Sci. 67 (2003) 4241-4248. Google Scholar

  • [2] A. Alonso, B. Simon: The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory 4 (1980) 251-270. Google Scholar

  • [3] N. I. Akhiezer: The Classical Moment Problem and Some Related Questions in Analysis, (Oliver & Boyd, Edinburgh, 1965). Google Scholar

  • [4] M. H. Annaby, Z. S. Mansour: On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankel transforms, Math. Proc. Camb. Phil. Soc. 147 (2009) 47-67. Web of ScienceGoogle Scholar

  • [5] B. M. Brown, J. S. Christiansen: On the Krein and Friedrichs extensions of a positive Jacobi operator, Expo. Math. 23 (2005) 179-186. Google Scholar

  • [6] G. Gasper, M. Rahman: Basic Hypergeometric Series, (Cambridge University Press, Cambridge, 1990). Google Scholar

  • [7] T. S. Chihara: An Introduction to Orthogonal Polynomials, (Gordon and Breach, Science Publishers, Inc., New York, 1978). Google Scholar

  • [8] T. Kato: Perturbation Theory for Linear Operators, (Springer-Verlag, Berlin, 1980). Google Scholar

  • [9] H. T. Koelink: Some basic Lommel polynomials, J. Approx. Theory 96 (1999) 345-365. Google Scholar

  • [10] H. T. Koelink,W. Van Assche: Orthogonal polynomials and Laurent polynomials related to the Hahn-Exton q-Bessel function, Constr. Approx. 11 (1995) 477-512. Google Scholar

  • [11] H. T. Koelink, R. F. Swarttouw: On the zeros of the Hahn-Exton q-Bessel function and associated q-Lommel polynomials, J. Math. Anal. Appl. 186 (1994) 690-710. Google Scholar

  • [12] L. O. Silva, R. Weder: On the two-spectra inverse problemfor semi-infinite Jacobi matrices in the limit-circle case,Math. Phys. Anal. Geom. 11 (2008) 131-154. Web of ScienceGoogle Scholar

  • [13] B. Simon: The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998) 82-203. Google Scholar

  • [14] F. Štampach, P. Šťovíček: The characteristic function for Jacobi matrices with applications, Linear Algebra Appl. 438 (2013) 4130-4155. Web of ScienceGoogle Scholar

  • [15] F. Štampach, P. Šťovíček: Special functions and spectrum of Jacobi matrices, Linear Algebra Appl., in press, available online: http://dx.doi.org/10.1016/j.laa.2013.06.024. CrossrefGoogle Scholar

  • [16] G. Teschl: Jacobi Operators and Completely Integrable Nonlinear Lattices, (AMS, Rhode Island, 2000). Google Scholar

  • [17] W. Van Assche: The ratio of q-like orthogonal polynomials, J. Math. Anal. Appl. 128 (1987) 535-547. Google Scholar

  • [18] J. Weidmann. Linear Operators in Hilbert Spaces. (Springer-Verlag, New York, 1980).Google Scholar

About the article

Received: 2014-02-05

Accepted: 2014-09-14

Published Online: 2014-10-01


Citation Information: Special Matrices, ISSN (Online) 2300-7451, DOI: https://doi.org/10.2478/spma-2014-0014.

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© 2014 F. Štampach, P. Šťovíček. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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