Titchmarsh-Weyl theory of the singular Hahn-Sturm-Liouville equation

In this work, we will consider the singular Hahn-Sturm-Liouville difference equation defined by -q-1D-ωq-1,q-1Dω,qy(x)+v(x)y(x)=λy(x), x∈(ω0,∞), where λ is a complex parameter, v is a real-valued continuous function at ω0 defined on [ω0,∞). These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the ω,q-Hahn difference operator Dω,q. We develop the ω,q-analogue of the classical Titchmarsh-Weyl theory for such equations. In other words, we study the existence of square-integrable solutions of the singular Hahn-Sturm-Liouville equation. Accordingly, first we define an appropriate Hilbert space in terms of Jackson-Norlund integral and then we study families of regular Hahn-Sturm-Liouville problems on [ω0,q-n], n∈N. Then we define a family of circles that converge either to a point or a circle. Thus, we will define the limit-point, limit-circle cases in the Hahn calculus setting by using Titchmarsh's technique.