Nagoya Mathematical Journal

q-Titchmarsh-Weyl theory: Series expansion

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

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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.

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

First available in Project Euclid: 1 March 2012

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


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.

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