Nagoya Mathematical Journal

$q$-Titchmarsh-Weyl theory: Series expansion

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

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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: 1 March 2012

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

Digital Object Identifier
doi:10.1215/00277630-1543787

Zentralblatt MATH identifier
06024988

Mathematical Reviews number (MathSciNet)
MR2891165

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 Mathematical Journal 205 (2012), 67--118. doi:10.1215/00277630-1543787. http://projecteuclid.org/euclid.nmj/1330611002.


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