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DOI: 10.46698/y9113-7002-9720-u

Titchmarsh-Weyl Theory of the Singular Hahn-Sturm-Liouville Equation

Allahverdiev, B. P. , Tuna, H.
Vladikavkaz Mathematical Journal 2021. Vol. 23. Issue 3.
Abstract:
In this work, we will consider the singular Hahn-Sturm-Liouville difference equation defined by \(-q^{-1}D_{-\omega q^{-1},q^{-1}}D_{\omega ,q}y( x) +v(x) y( x) =\lambda y(x)\), \(x\in (\omega _{0},\infty),\) where \(\lambda\) is a complex parameter, \(v\) is a real-valued continuous function at \(\omega _{0}\) defined on \([\omega _{0},\infty)\). These type equations are obtained when the ordinary derivative in the classical Sturm--Liouville problem is replaced by the \(\omega,q\)-Hahn difference operator \(D_{\omega,q}\). We develop the \(\omega,q\)-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 \([\omega_{0},q^{-n}]\), \(n\in \mathbb{N}\). 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.
Keywords: Hahn's Sturm-Liouville equation, limit-circle and limit-point cases, Titchmarsh-Weyl theory
Language: English Download the full text  
For citation: Allahverdiev, B. P. and Tuna, H. Titchmarsh-Weyl Theory of the Singular Hahn-Sturm-Liouville Equation, Vladikavkaz Math. J., 2021, vol. 23, no. 3, pp. 16-26. DOI 10.46698/y9113-7002-9720-u
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