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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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 + References 1. Weyl, H. Uber gewohnlicke Differentialgleichungen mit Singuritaten und die zugehorigen Entwicklungen willkurlicher Funktionen, Mathematische Annalen, 1910, vol. 68, no. 2, pp. 220-269. DOI: 10.1007/ BF01474161. 2. Titchmarsh, E. C. Eigenfunction Expansions Associated with Second-Order Differential Equations. Part I, 2nd Edition, Oxford, Clarendon Press, 1962. 3. Levitan, B. M. and Sargsjan, I. S. Sturm-Liouville and Dirac Operators, Mathematics and its Applications (Soviet Series), Dordrecht, Kluwer Academic Publishers Group, 1991. DOI: 10.1007/978-94-011-3748-5. 4. Yosida, K.On Titchmarsh-Kodaira's Formula Concerning Weyl-Stone's Eigenfunction Expansion, Nagoya Mathematical Journal, 1950, vol. 1, pp. 49-58. DOI: 10.1017/S0027763000022820. 5. Yosida, K. 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