Spectral properties of Sturm–Liouville operators on infinite metric graphs

• DOI: 10.1007/s13324-024-00937-8
• Publication Date: 2024-06-11T13:20:24Z
• Authors: Yihan Liu; Jun Yan; Jia Zhao
• Keywords: Infimum and supremum; Operator (biology); Spectral Properties; Quantum graph

This paper mainly deals with the Sturm-Liouville operator \begin{equation*} \mathbf{H}=\frac{1}{w(x)}\left( -\frac{\mathrm{d}}{\mathrm{d}x}p(x)\frac{ \mathrm{d}}{\mathrm{d}x}+q(x)\right) ,\text{ }x\in \Gamma \end{equation*} 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.