Eigenvalues of the Laplace Operator on Quantum Graphs Open Access

Abstract

This thesis focuses on estimates of eigenvalues on compact quantum graphs. On quantum graphs with all standard vertex condition, we prove an upper bound of eigenvalues based on the Davies inequality. We also prove some improvements of known upper bounds for eigenvalue gaps and ratios for metric trees with Dirichlet leaves. We finally establish a lower bound of eigenvalue gaps based on the idea of the weighted Cheeger constant on graphs with at least one Dirichlet vertex.

Table of Contents

Chapter 1: Introduction

Chapter 2: The upper bound of the spectral gap

Chapter 3: The upper bound for the tree

Chapter 4: Extensions of the upper bound

Chapter 5: The lower bound

About this Dissertation

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School	Laney Graduate School
Department	Mathematics
Degree	Ph.D.
Submission	Dissertation
Language	English
Research Field	Mathematics
Keyword	quantum graphs spectral theory
Committee Chair / Thesis Advisor	David Borthwick, Emory University
Committee Members	Yiran Wang, Emory University Maja Taskovic, Emory University

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