Investigation of the spectrum of singular Sturm–Liouville operators on unbounded time scales

Abstract

In this article, we consider the self-adjoint singular operators associated with the Sturm–Liouville expression $$\begin{aligned} Ly:=-\left[ p\left( t\right) y^{\Delta }\left( t\right) \right] ^{\nabla }+q\left( t\right) y\left( t\right) ,\ t\in (-\infty ,\infty )_{\mathbb {T}}. \end{aligned}$$on time scale $$\mathbb {T}$$. Some conditions are given for this operator to have a discrete spectrum. Further, we investigate the continuous spectrum of this operator. We also prove that the regular Sturm–Liouville operator on time scale is semi-bounded from below which is not studied in literature yet.