Diameter Bounds on the Spectral Gap of Quantum Graphs 公开

Luo Tianlang (Spring 2021)

Permanent URL: https://etd.library.emory.edu/concern/etds/zc77sr351?locale=zh
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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

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