Spectral properties of Sturm–Liouville operators on infinite metric graphs

Abstract

This paper mainly deals with the Sturm–Liouville operator

$$\begin{aligned} \textbf{H}=\frac{1}{w(x)}\left( -\frac{\textrm{d}}{\textrm{d}x}p(x)\frac{ \textrm{d}}{\textrm{d}x}+q(x)\right) ,\text { }x\in \Gamma \end{aligned}$$

acting in \(L_{w}^{2}\left( \Gamma \right) ,\) where \(\Gamma \) is a metric graph. We establish a relationship between the bottom of the spectrum and the positive solutions of quantum graphs, which is a generalization of the classical Allegretto–Piepenbrink theorem. Moreover, we prove the Persson-type theorem, which characterizes the infimum of the essential spectrum.