q-Titchmarsh-Weyl theory: Series expansion

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.