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Tytuł artykułu

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

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
2
Numer
1
Opis fizyczny
Daty
otrzymano
2014-02-05
zaakceptowano
2014-09-14
online
2014-10-01
Twórcy
autor
  • Department of Applied Mathematics, Faculty of Information Technology, Czech Technical University in Prague,
    Kolejní 2, 16000 Praha, Czech Republic
  • Department of Mathematics, Faculty of Nuclear Science, Czech Technical University in
    Prague, Trojanova 13, 12000 Praha, Czech Republic
Bibliografia
  • [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.
  • [2] A. Alonso, B. Simon: The Birman-Krein-Vishik theory of self-adjoint extensions of semibounded operators, J. Operator Theory4 (1980) 251-270.
  • [3] N. I. Akhiezer: The Classical Moment Problem and Some Related Questions in Analysis, (Oliver & Boyd, Edinburgh, 1965).
  • [4] M. H. Annaby, Z. S. Mansour: On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankeltransforms, Math. Proc. Camb. Phil. Soc. 147 (2009) 47-67.[WoS]
  • [5] B. M. Brown, J. S. Christiansen: On the Krein and Friedrichs extensions of a positive Jacobi operator, Expo. Math. 23 (2005)179-186.
  • [6] G. Gasper, M. Rahman: Basic Hypergeometric Series, (Cambridge University Press, Cambridge, 1990).
  • [7] T. S. Chihara: An Introduction to Orthogonal Polynomials, (Gordon and Breach, Science Publishers, Inc., New York, 1978).
  • [8] T. Kato: Perturbation Theory for Linear Operators, (Springer-Verlag, Berlin, 1980).
  • [9] H. T. Koelink: Some basic Lommel polynomials, J. Approx. Theory 96 (1999) 345-365.
  • [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.
  • [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.
  • [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]
  • [13] B. Simon: The classical moment problem as a self-adjoint finite difference operator, Adv. Math. 137 (1998) 82-203.
  • [14] F. Štampach, P. Šťovíček: The characteristic function for Jacobi matrices with applications, Linear Algebra Appl. 438 (2013)4130-4155.[WoS]
  • [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]
  • [16] G. Teschl: Jacobi Operators and Completely Integrable Nonlinear Lattices, (AMS, Rhode Island, 2000).
  • [17] W. Van Assche: The ratio of q-like orthogonal polynomials, J. Math. Anal. Appl. 128 (1987) 535-547.
  • [18] J. Weidmann. Linear Operators in Hilbert Spaces. (Springer-Verlag, New York, 1980).
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_spma-2014-0014
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