Diameter Bounds on the Spectral Gap of Quantum Graphs Open Access

Abstract

A quantum graph is a metric graph equipped with a differential operator. The spectrum of a compact quantum graph is real and discrete, where its first positive eigenvalue is referred to as the spectral gap. We present a method to study the upper bound on the spectral gap of quantum graphs in terms of the diameter (and possibly of the total length and the total number of vertices) by reduction to a Sturm-Liouville problem.

Table of Contents

1 Introduction 1

1.1 Background on quantum graphs . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2 Spectral theory of quantum graphs . . . . . . . . . . . . . . . . . . . . . . . 1

1.3 Summary of results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2 Rayleigh Quotient 4

3 Surgery on Quantum Graphs 7

4 Reduction to One-dimensional Problem 14

5 Application: An Estimate in terms of Diameter and Total Length 21

About this Honors Thesis

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School	Emory College
Department	Mathematics
Degree	B.S.
Submission	Honors Thesis
Language	English
Research Field	Mathematics
Keyword	Mathematical Physics Spectral Theory
Committee Chair / Thesis Advisor	David Borthwick, Emory University
Committee Members	Maja Taskovic, Emory University Effrosyni Seitaridou, Emory University

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