Теория Титчмарша - Вейля сингулярного уравнения Хана - Штурма - Лиувилля

In this work, we will consider the singularHahn--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 acomplex parameter, $v$ is a real-valued continuous function at$\omega _{0}$ defined on $[\omega _{0},\infty)$. These typeequations are obtained when the ordinary derivative in the classicalSturm--Liouville problem is replaced by the $\omega,q$-Hahndifference operator $D_{\omega,q}$. We develop the $\omega,q$-analogue of the classicalTitchmarsh--Weyl theory for such equations. In other words, we study the existence ofsquare-integrable solutions of the singular Hahn--Sturm--Liouvilleequation. Accordingly, first we define an appropriate Hilbertspace in terms of Jackson--N\"{o}rlund integral and then we studyfamilies of regular Hahn--Sturm--Liouville problems on$[\omega_{0},q^{-n}]$, $n\in \mathbb{N}$. Then we define a family ofcircles that converge either to a point or a circle. Thus, we willdefine the limit-point, limit-circle cases in the Hahn calculussetting by using Titchmarsh's technique.