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Spectral properties of Sturm-Liouville operators on infinite metric graphs

Yihan Liu, Jun Yan, Jia Zhao

TL;DR

This work extends spectral theory for Sturm–Liouville operators to infinite metric graphs with Kirchhoff conditions by proving an Allegretto–Piepenbrink-type theorem that links the bottom of the spectrum to the existence of positive solutions of $l y=\lambda y$, and a Persson-type theorem that characterizes the essential spectrum via exhaustion by finite subgraphs. The approach centers on quadratic forms $\mathbf{t}_{q}^{0}=\mathbf{t}_{0}^{0}+\mathbf{q}$, their closability, and the associated self-adjoint realization $\mathbf{H}_{\mathbf{t}_{q}}$, with a Dirichlet realization $\mathbf{H}_{\mathrm{D}}$ playing a key role. The results rely on finite-subgraph approximations, Harnack-type estimates, and compactness arguments to pass to the infinite graph, yielding precise spectral thresholds that have implications for parabolic and quantum-graph dynamics on networks. Overall, the paper provides sharp connections between spectral data and positive solutions on metric graphs under concrete coefficient and geometry conditions.

Abstract

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 Γ\end{equation*} acting in $L_{w}^{2}\left( Γ\right) ,$ where $Γ$ 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.

Spectral properties of Sturm-Liouville operators on infinite metric graphs

TL;DR

This work extends spectral theory for Sturm–Liouville operators to infinite metric graphs with Kirchhoff conditions by proving an Allegretto–Piepenbrink-type theorem that links the bottom of the spectrum to the existence of positive solutions of , and a Persson-type theorem that characterizes the essential spectrum via exhaustion by finite subgraphs. The approach centers on quadratic forms , their closability, and the associated self-adjoint realization , with a Dirichlet realization playing a key role. The results rely on finite-subgraph approximations, Harnack-type estimates, and compactness arguments to pass to the infinite graph, yielding precise spectral thresholds that have implications for parabolic and quantum-graph dynamics on networks. Overall, the paper provides sharp connections between spectral data and positive solutions on metric graphs under concrete coefficient and geometry conditions.

Abstract

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 Γ\end{equation*} acting in where 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.
Paper Structure (4 sections, 6 theorems, 86 equations)

This paper contains 4 sections, 6 theorems, 86 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on metric graphs
  3. Allegretto-Piepenbrink-type theorem on metric graphs
  4. Persson-type theorem on metric graphs

Key Result

Theorem 1

For any $\lambda \in \mathbb{R}$, $(1)$ if there exists a positive solution $y>0$ on $\Gamma$ for $ly=\lambda y$, then $\inf \sigma (\mathbf{H}_{\mathbf{t}_{q}})\geq \lambda ;$$\space(2)$ if $\inf \sigma (\mathbf{H}_{\mathbf{t}_{q}})>\lambda$, then there exists a positive solution $y>0$ on $\Gamma$

Theorems & Definitions (17)

  • Theorem 1: Allegretto-Piepenbrink-type theorem
  • proof
  • Remark 1
  • Theorem 2: Persson-type theorem
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Remark 2
  • Lemma 3
  • ...and 7 more