The Laplace and Heat Operators on Quantum Graphs Open Access

Jones, Kenny (Spring 2022)

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This thesis analyzes the Laplace and heat operators on quantum graphs. The thesis is separated into four chapters, 

Chapter 1: Introduction to quantum graphs, the Laplace operator, and summary of the main results.

Chapter 2: Strategies for bounding the spectral gap of a quantum graph, including a sharp upper bound for the spectral gap using the diameter and total number of vertices as parameters.

Chapter 3: Bounds for the heat kernel of a quantum graph. The main results include a bound for small time and identifying a class of edges that can be bound by a Neumann interval. 

Chapter 4: Finds two mean value formulas for the heat equation on a quantum graph. Proves an additional bound for the mean value formula using the one dimensional free heat kernel.

Table of Contents

Chapter 1.

Introduction 7

1. Motivation 7

2. Introduction to Quantum Graphs 7

3. The Laplacian and Its Spectrum 9

4. Main Results 11

Chapter 2. The Spectral Gap 13

1. Introduction to the Spectral Gap 13

2. Diameter Bounds and Pumpkin Chains 18

3. Sharp Diameter Bound for Quantum Graphs 22

Chapter 3. The Heat Kernel 27

1. Introduction to the Heat Kernel 27

2. Bond Scattering Matrix and Heat Kernel Formula 28

3. Heat Kernel for Small Time 29

4. Construction of the Graph G∗ n 30

5. Identifying Paths along G′ and G∗ n 33

6. Bounding Coefficients in the Heat Kernel Formula 38

7. Direct Path Bounded Edges 43

8. Comparison Between Neumann Interval and Direct Path Bounded Edges 47

9. Neumann Comparison: Off Diagonal Results 50

Chapter 4. Mean Value Theorem 53

1. Mean Value Formula 53

2. Bounding the Mean Value Formula 55

Bibliography 61

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