**Contemporary Mathematics**

Volume: 415;
2006;
307 pp;
Softcover

MSC: Primary 05; 11; 15; 28; 34; 35; 58; 60; 81; 92; 94;

Print ISBN: 978-0-8218-3765-8

Product Code: CONM/415

List Price: $97.00

AMS Member Price: $77.60

MAA Member Price: $87.30

**Electronic ISBN: 978-0-8218-8094-4
Product Code: CONM/415.E**

List Price: $97.00

AMS Member Price: $77.60

MAA Member Price: $87.30

# Quantum Graphs and Their Applications

Share this page *Edited by *
*Gregory Berkolaiko; Robert Carlson; Stephen A. Fulling; Peter Kuchment*

This volume is a collection of articles dedicated to quantum graphs, a newly emerging interdisciplinary field related to various areas of mathematics and physics. The reader can find a broad overview of the theory of quantum graphs. The articles present methods coming from different areas of mathematics: number theory, combinatorics, mathematical physics, differential equations, spectral theory, global analysis, and theory of fractals. They also address various important applications, such as Anderson localization, electrical networks, quantum chaos, mesoscopic physics, superconductivity, optics, and biological modeling.

#### Readership

Graduate students and research mathematicians interested in quantum graphs and application to number theory, mathematical physics, and other areas.

# Table of Contents

## Quantum Graphs and Their Applications

- Contents v6 free
- Preface vii8 free
- Fluctuation-Based Proof of the Stability of AC Spectra of Random Operators on Thee Graphs 112 free
- Metrized Graphs, Laplacian Operators, and Electrical Networks 1526
- Form Factor Expansion for Large Graphs: a Diagrammatic Approach 3546
- The Spectral Form Factor for Quantum Graphs with Spin-Orbit Coupling 5162
- Linear Network Models Related to Blood Flow 6576
- Localization on Avron-Exner-Last Graphs: I. Local Perturbations 8192
- Weighted Laplacians and the Sigma Function of a Graph 93104
- Approximations of Permutation-Symmetric Vertex Couplings in Quantum Graphs 109120
- Resistance of Random Sierpiński Gaskets 121132
- 1. Introduction 121132
- 2. Energy forms and effective resistance 122133
- 3. Parametrization of energy forms on the Sierpiński gasket 124135
- 4. Distribution of energy and resistances 127138
- 5. Harmonic functions on random Sierpiński gaskets 131142
- 6. Approximation by quantum graphs 133144
- References 134145

- Small Diffusion Asymptotics for Exit Problems on Graphs 137148
- Determinant of the Schrödinger Operator on a Metric Graph 151162
- Local Spectral Density and Vacuum Energy Near a Quantum Graph Vertex 161172
- What Are Zeta Functions of Graphs and What Are They Good For? 173184
- Fluctuation Statistics For Quantum Star Graphs 191202
- Laplacians on Metric Graphs: Eigenvalues, Resolvents and Semigroups 201212
- Transition From a Network of Thin Fibers to the Quantum Graph: an Explicitly Solvable Model 227238
- On the Limiting Absorption Principle and Spectra of Quantum Graphs 241252
- Quantum Mechanics, Superconductivity and Fluid Flow in Narrow Networks 251262
- A Relation Between the Bond Scattering Matrix and the Spectral Counting Function for Quantum Graphs 269280
- The Quantum Graph as a Limit of a Network of Physical Wires 283294
- On the Trace Formula for Quantum Star Graphs 293304