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q-Titchmarsh-Weyl theory: series expansion

Published online by Cambridge University Press:  11 January 2016

M. H. Annaby ,
Z. S. Mansour  and
I. A. Soliman
M. H. Annaby
Affiliation:
Department of Mathematics, and statistics and Physics Qatar University, Doha Qatarmannaby@qu.edu.qa
Z. S. Mansour
Affiliation:
Department of Mathematics King Saudi University, Riyadh Riyadh 11451, Kingdom of Saudi Arabiazeinabs98@hotmail.com
I. A. Soliman
Affiliation:
Department of Mathematics Cairo University, Giza Egypteman.a.raouf@hotmail.com
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Abstract

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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.

Keywords

34B2434L1039A13
Type
Research Article
Information
Nagoya Mathematical Journal , Volume 205 , March 2012 , pp. 67 - 118
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 2012

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