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

F. Štampach; P. Šťovíček

Special Matrices (2014)

  • Volume: 2, Issue: 1, page 131-147, electronic only
  • ISSN: 2300-7451

Abstract

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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.

How to cite

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F. Štampach, and P. Šťovíček. "The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix." Special Matrices 2.1 (2014): 131-147, electronic only. <http://eudml.org/doc/266893>.

@article{F2014,
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.},
author = {F. Štampach, P. Šťovíček},
journal = {Special Matrices},
keywords = {Jacobi matrix; Hahn-Exton q-Bessel function; self-adjoint extension; spectral problem; Hahn-Exton -Bessel function},
language = {eng},
number = {1},
pages = {131-147, electronic only},
title = {The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix},
url = {http://eudml.org/doc/266893},
volume = {2},
year = {2014},
}

TY - JOUR
AU - F. Štampach
AU - P. Šťovíček
TI - The Hahn-Exton q-Bessel function as the characteristic function of a Jacobi matrix
JO - Special Matrices
PY - 2014
VL - 2
IS - 1
SP - 131
EP - 147, electronic only
AB - 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.
LA - eng
KW - Jacobi matrix; Hahn-Exton q-Bessel function; self-adjoint extension; spectral problem; Hahn-Exton -Bessel function
UR - http://eudml.org/doc/266893
ER -

References

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  1. [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. Zbl1063.33023
  2. [2] A. Alonso, B. Simon: The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory 4 (1980) 251-270. Zbl0467.47017
  3. [3] N. I. Akhiezer: The Classical Moment Problem and Some Related Questions in Analysis, (Oliver & Boyd, Edinburgh, 1965). 
  4. [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. [WoS] Zbl1175.33013
  5. [5] B. M. Brown, J. S. Christiansen: On the Krein and Friedrichs extensions of a positive Jacobi operator, Expo. Math. 23 (2005) 179-186. Zbl1078.39019
  6. [6] G. Gasper, M. Rahman: Basic Hypergeometric Series, (Cambridge University Press, Cambridge, 1990). 
  7. [7] T. S. Chihara: An Introduction to Orthogonal Polynomials, (Gordon and Breach, Science Publishers, Inc., New York, 1978). 
  8. [8] T. Kato: Perturbation Theory for Linear Operators, (Springer-Verlag, Berlin, 1980). Zbl0435.47001
  9. [9] H. T. Koelink: Some basic Lommel polynomials, J. Approx. Theory 96 (1999) 345-365. Zbl0926.33003
  10. [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. Zbl0855.33011
  11. [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. Zbl0811.33013
  12. [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. [WoS] Zbl1189.47030
  13. [13] B. Simon: The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998) 82-203. Zbl0910.44004
  14. [14] F. Štampach, P. Šťovíček: The characteristic function for Jacobi matrices with applications, Linear Algebra Appl. 438 (2013) 4130-4155. [WoS] Zbl1300.47040
  15. [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. [Crossref] Zbl1330.47033
  16. [16] G. Teschl: Jacobi Operators and Completely Integrable Nonlinear Lattices, (AMS, Rhode Island, 2000). 
  17. [17] W. Van Assche: The ratio of q-like orthogonal polynomials, J. Math. Anal. Appl. 128 (1987) 535-547. Zbl0634.33018
  18. [18] J. Weidmann. Linear Operators in Hilbert Spaces. (Springer-Verlag, New York, 1980). Zbl0434.47001

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