Geodesic
Titchmarsh–Weyl theory of the singular Hahn–Sturm–Liouville equation
Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 3, pp. 16-26 Cet article a éte moissonné depuis la source Math-Net.Ru

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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–Nörlund 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. 
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     title = {Titchmarsh{\textendash}Weyl theory of the singular {Hahn{\textendash}Sturm{\textendash}Liouville} equation},
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B. P. Allahverdiev; H. Tuna. Titchmarsh–Weyl theory of the singular Hahn–Sturm–Liouville equation. Vladikavkazskij matematičeskij žurnal, Tome 23 (2021) no. 3, pp. 16-26. http://geodesic.mathdoc.fr/item/VMJ_2021_23_3_a1/

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