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

\(q\)-Titchmarsh-Weyl theory: series expansion. (English) Zbl 1266.34037

Nagoya Math. J. 205, 67-118 (2012).
The authors establish a \(q\)-Titchmarsh-Weyl theory for singular \(q\)-Sturm-Liouville problems. The theory for singular Sturm-Liouville problems of the type \[ \begin{aligned} -y'' + \nu(x) y = \lambda y,&\quad 0\leq x < +\infty, \\ \cos \alpha y(0) + \sin \alpha y'(0) = 0, &\end{aligned} \] has been established by Weyl. The goal of this paper is to establish a corresponding theory for singular Sturm-Liouville \(q\)-difference operators when the derivative is replaced by Jackson’s \(q\)-difference operator \(D_{q}\). The \(q\)-limit-point and \(q\)-limit circle singularities are defined, sufficient conditions which guarantee that the singular point is in a limit-point case are given. The resolvent is constructed in terms of Green’s function of the problem. Moreover, the eigenfunction expansion in its series form is derived. Finally, a detailed example involving Jackson \(q\)-Bessel functions is given, which leads to the completeness of a wide class of \(q\)-cylindrical functions.
Reviewer: Petr Necesal (Plzen)

Cited in 19 Documents

MSC:

34B24 Sturm-Liouville theory
34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
39A12 Discrete version of topics in analysis

Keywords:

\(q\)-Sturm-Liouville problem; \(q\)-limit-point; \(q\)-limit-circle; \(q\)-Bessel functions; \(q\)-cylindrical functions
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References:

[1] L. D. Abreu, A q-sampling theorem related to the q-Hankel transform , Proc. Amer. Math. Soc. 133 (2004), 1197-1203. · Zbl 1062.33015 · doi:10.1090/S0002-9939-04-07589-6
[2] L. D. Abreu, Functions q-orthogonal with respect to their own zeros , Proc. Amer. Math. Soc. 134 (2006), 2695-2701. · Zbl 1091.33013 · doi:10.1090/S0002-9939-06-08285-2
[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. · Zbl 1219.33014 · doi:10.1007/0-387-24233-3_2
[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. · Zbl 1063.33023 · doi:10.1155/S016117120320613X
[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. · Zbl 1143.39009 · doi:10.4171/ZAA/1338
[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. · Zbl 0920.33001
[8] M. H. Annaby, q-Type sampling theorems , Results Math. 44 (2003), 214-225. · Zbl 1046.94002 · doi:10.1007/BF03322983
[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. · Zbl 1073.33012 · doi:10.1088/0305-4470/38/17/005
[10] M. H. Annaby and Z. S. Mansour, On the zeros of basic finite Hankel transforms , J. Math. Anal. Appl. 323 (2006), 1091-1103. · Zbl 1108.44002 · doi:10.1016/j.jmaa.2005.11.020
[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. · Zbl 1175.33013 · doi:10.1017/S0305004109002357
[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. · Zbl 1219.39003 · doi:10.1002/mana.200810037
[13] J. M. Berezanskii, Expansions in Eigenfunctions of Selfadjoint Operators , Amer. Math. Soc., Providence, 1968. · Zbl 0157.16601
[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. · Zbl 1173.47018 · doi:10.1007/s00220-008-0623-1
[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. · Zbl 0860.33012 · doi:10.1017/S0305004100073916
[16] J. Bustoz and J. L. Cardoso, Basic analog of Fourier series on a q-linear grid , J. Approx. Theory 112 (2001), 134-157. · Zbl 1077.33030 · doi:10.1006/jath.2001.3599
[17] J. Bustoz and S. K. Suslov, Basic analog of Fourier series on a q-quadratic grid , Methods Appl. Anal. 5 (1998), 1-38. · Zbl 0961.33013
[18] R. D. Carmichael, The general theory of linear q-difference equations , Amer. J. Math. 34 (1912), 147-168. · JFM 43.0411.02 · doi:10.2307/2369887
[19] R. D. Carmichael, Linear difference equations and their analytic solutions , Trans. Amer. Math. Soc. 12 , no. 1 (1911), 99-134. · JFM 42.0359.01 · doi:10.1090/S0002-9947-1911-1500883-6
[20] R. D. Carmichael, On the theory of linear difference equations , Amer. J. Math. 35 (1913), 163-182. · JFM 44.0397.03 · doi:10.2307/2370279
[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. · Zbl 1089.33016 · doi:10.1090/S0002-9947-05-03785-2
[22] R. Conway, Functions in One Complex Variable , Springer, New York, 1999.
[23] C. T. Fulton, Parametrizations of Titchmarsh’s m ( \lambda ) -functions in the limit circle case , Trans. Amer. Math. Soc. 229 (1977), 51-63. · Zbl 0358.34021 · doi:10.2307/1998499
[24] W. Hahn, Beiträge zur Theorie der Heineschen Reihen , Math. Nachr. 2 (1949), 340-379. · Zbl 0033.05703 · doi:10.1002/mana.19490020604
[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. · Zbl 0807.34016 · doi:10.1137/1034117
[26] M. E. H. Ismail, The zeros of basic Bessel functions, the functions J \nu + ax ( x ) and associated orthogonal polynomials , J. Math. Anal. Appl. 86 (1982), 1-19. · Zbl 0483.33004 · doi:10.1016/0022-247X(82)90248-7
[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. · JFM 35.0487.02
[30] F. H. Jackson, On q-definite integrals , Quart. J. Pure Appl. Math. 41 (1910), 193-203. · JFM 41.0317.04
[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. · Zbl 0811.33013 · doi:10.1006/jmaa.1994.1327
[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. · Zbl 0759.33007 · doi:10.2307/2154118
[33] S. Lang, Complex Analysis , Springer, New York, 1992. · Zbl 0766.34030
[34] A. Lavagno, Basic-deformed quantum mechanics , Rep. Math. Phys. 64 (2009), 79-91. · Zbl 1204.81095 · doi:10.1016/S0034-4877(09)90021-0
[35] N. Levinson, A simplified proof of the expansion theorem for singular second order linear differential equations , Duke Math. J. 18 (1951), 57-71. · Zbl 0044.31302 · doi:10.1215/S0012-7094-51-01806-6
[36] B. M. Levitan and I. S. Sargsjan, Introduction to Spectral Theory: Selfadjoint Ordinary Differential Operators , Amer. Math. Soc., Providence, 1975. · Zbl 0302.47036
[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. · Zbl 0776.17014 · doi:10.1007/BF02096769
[39] M. A. Naimark, Linear Differential Operators, Part II: Lineal Differential Operators in Hilbert Space , Frederick Ungar, New York, 1968. · Zbl 0227.34020
[40] M. H. Stone, A comparison of the series of Fourier and Birkhoff , Trans. Amer. Math. Soc. 28 (1926), 695-761. · JFM 52.0456.02 · doi:10.1090/S0002-9947-1926-1501372-6
[41] M. H. Stone, Linear Transformations in Hilbert Space and Their Application to Analysis , Amer. Math. Soc., Providence, 1932. · Zbl 0005.40003
[42] S. K. Suslov, Some expansions in basic Fourier series and related topics , J. Approx. Theory 115 (2002), 289-353. · Zbl 1009.42020 · doi:10.1006/jath.2001.3659
[43] S. K. Suslov, An Introduction to Basic Fourier Series , Kluwer Ser. Dev. Math. 9 , Kluwer Academic, Dordrecht, 2003. · Zbl 1038.42001
[44] R. F. Swarttouw, The Hahn-Exton q-Bessel function , Ph.D. dissertation, Technical University of Delft, Delft, Netherlands, 1992. · Zbl 0759.33008
[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. · Zbl 0822.33009 · doi:10.2307/2160480
[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. · Zbl 0031.30801 · doi:10.4153/CJM-1949-018-x
[47] E. Titchmarsh, Eigenfunction Expansions Associated with Second-Order Differential Equations, I , 2nd ed., Clarendon Press, Oxford, 1962. · Zbl 0099.05201
[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. · JFM 41.0344.01
[50] H. Weyl, Über Gewöhnliche differentialgleichungen mit singularitäten und die zugehörigen entwicklung willkürlicher funktionen , Math. Ann. 68 (1910), 220-269. · JFM 41.0343.01
[51] K. Yosida, On Titchmarsh-Kodaira’s formula concerning Weyl-Stone’s eingenfunction expansion , Nagoya Math. J. 1 (1950), 49-58. · Zbl 0038.24802
[52] K. Yosida, Lectures on Differential and Integral Equations , Springer, New York, 1960. · Zbl 0090.08401
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