Nagoya Mathematical Journal

$q$-Titchmarsh-Weyl theory: Series expansion

M. H. Annaby, Z. S. Mansour, and I. A. Soliman

Full-text: Open access

Abstract

We establish a $q$-Titchmarsh-Weyl theory for singular $q$-Sturm-Liouville problems. We define $q$-limit-point and $q$-limit circle singularities, and we give sufficient conditions which guarantee that the singular point is in a limit-point case. The resolvent is constructed in terms of Green’s function of the problem. We derive the eigenfunction expansion in its series form. A detailed worked example involving Jackson $q$-Bessel functions is given. This example leads to the completeness of a wide class of $q$-cylindrical functions.

Article information

Source
Nagoya Math. J. Volume 205 (2012), 67-118.

Dates
First available in Project Euclid: 1 March 2012

Permanent link to this document
http://projecteuclid.org/euclid.nmj/1330611002

Digital Object Identifier
doi:10.1215/00277630-1543787

Mathematical Reviews number (MathSciNet)
MR2891165

Zentralblatt MATH identifier
06024988

Subjects
Primary: 34B24: Sturm-Liouville theory [See also 34Lxx] 34L10: Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions 39A13: Difference equations, scaling ($q$-differences) [See also 33Dxx]

Citation

Annaby, M. H.; Mansour, Z. S.; Soliman, I. A. q -Titchmarsh-Weyl theory: Series expansion. Nagoya Math. J. 205 (2012), 67--118. doi:10.1215/00277630-1543787. http://projecteuclid.org/euclid.nmj/1330611002.


Export citation

References

  • [1] L. D. Abreu, A q-sampling theorem related to the q-Hankel transform, Proc. Amer. Math. Soc. 133 (2004), 1197–1203.
  • [2] L. D. Abreu, Functions q-orthogonal with respect to their own zeros, Proc. Amer. Math. Soc. 134 (2006), 2695–2701.
  • [3] L. D. Abreu and J. Bustoz, “On the completeness of sets of q-Bessel functions” in Theory and Applications of Special Functions, Dev. Math. 13, Springer, New York, 2005, 29–38.
  • [4] L. D. Abreu, J. Bustoz, and J. L. Caradoso, The roots of the third Jackson q-Bessel functions, Int. J. Math. Math. Sci. 67 (2003), 4241–4248.
  • [5] M. H. Abu-Risha, M. H. Annaby, M. E. H. Ismail, and Z. S. Mansour, Linear q-difference equations, Z. Anal. Anwend. 26 (2007), 481–494.
  • [6] L. Ahlfors, Complex Analysis, Cambridge University Press, Cambridge, 1999.
  • [7] G. E. Andrews, R. Askey, and R. Roy, Special Functions, Cambridge University Press, Cambridge, 1999.
  • [8] M. H. Annaby, q-Type sampling theorems, Results Math. 44 (2003), 214–225.
  • [9] M. H. Annaby and Z. S. Mansour, Basic Sturm-Liouville problems, J. Phys. A 38 (2005), 3775–3797; Correction, J. Phys. A 39 (2006), 8747.
  • [10] M. H. Annaby and Z. S. Mansour, On the zeros of basic finite Hankel transforms, J. Math. Anal. Appl. 323 (2006), 1091–1103.
  • [11] M. H. Annaby and Z. S. Mansour, On the zeros of the second and third Jackson q-Bessel functions and their associated q-Hankel transforms, Math. Proc. Cambridge Philos. Soc. 147 (2009), 47–67.
  • [12] M. H. Annaby and Z. S. Mansour, Asymptotic formulae for eigenvalues and eigenfunctions of q-Sturm-Liouville problems, Math. Nachr. 284 (2011), 443–470.
  • [13] J. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators, Amer. Math. Soc., Providence, 1968.
  • [14] B. M. Brown, J. S. Christiansen, and K. M. Schmidt, Spectral properties of a q-Sturm-Liouville operator, Comm. Math. Phys. 287 (2009), 259–274.
  • [15] B. M. Brown, W. D. Evans, and M. E. H. Ismail, The Askey-Wilson polynomials and q-Sturm-Liouville problems, Math. Proc. Cambridge Philos. Soc. 119 (1996), 1–16.
  • [16] J. Bustoz and J. L. Cardoso, Basic analog of Fourier series on a q-linear grid, J. Approx. Theory 112 (2001), 134–157.
  • [17] J. Bustoz and S. K. Suslov, Basic analog of Fourier series on a q-quadratic grid, Methods Appl. Anal. 5 (1998), 1–38.
  • [18] R. D. Carmichael, The general theory of linear q-difference equations, Amer. J. Math. 34 (1912), 147–168.
  • [19] R. D. Carmichael, Linear difference equations and their analytic solutions, Trans. Amer. Math. Soc. 12, no. 1 (1911), 99–134.
  • [20] R. D. Carmichael, On the theory of linear difference equations, Amer. J. Math. 35 (1913), 163–182.
  • [21] J. S. Christiansen and M. E. H. Ismail, A moment problem and a family of integral evaluations, Trans. Amer. Math. Soc. 358, no. 9 (2006), 4071–4097.
  • [22] R. Conway, Functions in One Complex Variable, Springer, New York, 1999.
  • [23] C. T. Fulton, Parametrizations of Titchmarsh’s m(λ)-functions in the limit circle case, Trans. Amer. Math. Soc. 229 (1977), 51–63.
  • [24] W. Hahn, Beiträge zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949), 340–379.
  • [25] M. Hajmirzaahmad and A. M. Krall, Singular second-order operators: The maximal and minimal operators, and selfadjoint operators in between, SIAM Rev. 34 (1992), 614–634.
  • [26] M. E. H. Ismail, The zeros of basic Bessel functions, the functions Jν+ax(x) and associated orthogonal polynomials, J. Math. Anal. Appl. 86 (1982), 1–19.
  • [27] M. E. H. Ismail, Classical and Quantum Orthogonal Polynomials in One Variable, Encyclopedia Math. Appl. 98, Cambridge University Press, Cambridge, 2005.
  • [28] M. E. H. Ismail, On Jackson’s third q-Bessel function, preprint, 1996.
  • [29] F. H. Jackson, The applications of basic numbers to Bessel’s and Legendre’s equations, Proc. Lond. Math. Soc. (3) 2 (1905), 192–220.
  • [30] F. H. Jackson, On q-definite integrals, Quart. J. Pure Appl. Math. 41 (1910), 193–203.
  • [31] H. T. Koelink and 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.
  • [32] T. H. Koornwinder and R. F. Swarttouw, On a q-analog of the Fourier and Hankel transforms, Trans. Amer. Math. Soc. 333 (1992), no. 1, 445–461.
  • [33] S. Lang, Complex Analysis, Springer, New York, 1992.
  • [34] A. Lavagno, Basic-deformed quantum mechanics, Rep. Math. Phys. 64 (2009), 79–91.
  • [35] N. Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations, Duke Math. J. 18 (1951), 57–71.
  • [36] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators, Amer. Math. Soc., Providence, 1975.
  • [37] B. M. Levitan and I. S. Sargsjan, Sturm Liouville and Dirac Operators, Kluwer, Dordrecht, 1991.
  • [38] A. Matsuo, Jackson integrals of Jordan-Pochhammer type and quantum Knizhnik-Zamolodchikov equations, Comm. Math. Phys. 151 (1993), 263–273.
  • [39] M. A. Naimark, Linear Differential Operators, Part II: Lineal Differential Operators in Hilbert Space, Frederick Ungar, New York, 1968.
  • [40] M. H. Stone, A comparison of the series of Fourier and Birkhoff, Trans. Amer. Math. Soc. 28 (1926), 695–761.
  • [41] M. H. Stone, Linear Transformations in Hilbert Space and Their Application to Analysis, Amer. Math. Soc., Providence, 1932.
  • [42] S. K. Suslov, Some expansions in basic Fourier series and related topics, J. Approx. Theory 115 (2002), 289–353.
  • [43] S. K. Suslov, An Introduction to Basic Fourier Series, Kluwer Ser. Dev. Math. 9, Kluwer Academic, Dordrecht, 2003.
  • [44] R. F. Swarttouw, The Hahn-Exton q-Bessel function, Ph.D. dissertation, Technical University of Delft, Delft, Netherlands, 1992.
  • [45] R. F. Swarttouw and H. G. Meijer, A q-analogue of the Wronskian and a second solution of the Hahn-Exton q-Bessel difference equation, Proc. Amer. Math. Soc. 120 (1994), 855–864.
  • [46] E. C. Titchmarsh, On the uniqueness of the Green’s function associated with a second-order differential equation, Canad. J. Math. 1 (1949), 191–198.
  • [47] E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, I, 2nd ed., Clarendon Press, Oxford, 1962.
  • [48] H. Weyl, Gewöhnliche linear differentialgleichungen mit singularitäten stellen und ihre eigenfunktionen, Göttingen Ges. Wiss. Nach. 68 (1909), 73–64.
  • [49] H. Weyl, Gewöhnliche linear differentialgleichungen mit singularitäten stellen und ihre eigenfunktionen, Göttingen Ges. Wiss. Nach. 68 (1910), 442–467.
  • [50] H. Weyl, Über Gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklung willkürlicher funktionen, Math. Ann. 68 (1910), 220–269.
  • [51] K. Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eingenfunction expansion, Nagoya Math. J. 1 (1950), 49–58.
  • [52] K. Yosida, Lectures on Differential and Integral Equations, Springer, New York, 1960.