**Mathematical Surveys and Monographs**

Volume: 186;
2013;
270 pp;
Hardcover

MSC: Primary 34; 35; 81; 58; 05; 82; 47;

Print ISBN: 978-0-8218-9211-4

Product Code: SURV/186

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

**Electronic ISBN: 978-0-8218-9455-2
Product Code: SURV/186.E**

List Price: $86.00

AMS Member Price: $68.80

MAA Member Price: $77.40

#### Supplemental Materials

# Introduction to Quantum Graphs

Share this page
*Gregory Berkolaiko; Peter Kuchment*

A “quantum graph” is a graph
considered as a one-dimensional complex and equipped with a
differential operator (“Hamiltonian”).
Quantum graphs arise naturally as simplified
models in mathematics, physics, chemistry, and engineering when one
considers propagation of waves of various nature through a
quasi-one-dimensional (e.g., “meso-” or
“nano-scale”) system that looks like a thin neighborhood
of a graph. Works that currently would be classified as discussing
quantum graphs have been appearing since at least the 1930s, and since
then, quantum graphs techniques have been applied successfully in
various areas of mathematical physics, mathematics in general and its
applications. One can mention, for instance, dynamical systems theory,
control theory, quantum chaos, Anderson localization,
microelectronics, photonic crystals, physical chemistry,
nano-sciences, superconductivity theory, etc.

Quantum graphs present many non-trivial mathematical challenges,
which makes them dear to a mathematician's heart. Work on quantum
graphs has brought together tools and intuition coming from graph
theory, combinatorics, mathematical physics, PDEs, and spectral
theory.

This book provides a comprehensive introduction to the topic,
collecting the main notions and techniques. It also contains a survey
of the current state of the quantum graph research and
applications.

#### Readership

Graduate students and research mathematicians interested in this new area of mathematics and its applications.

#### Table of Contents

# Table of Contents

## Introduction to Quantum Graphs

- Cover Cover11 free
- Title page iii4 free
- Contents vii8 free
- Preface xi12 free
- Introduction xiii14 free
- Operators on graphs. Quantum graphs 116 free
- Quantum graphs–Some special topics 3752
- Spectra of quantum graphs 6580
- Spectra of periodic graphs 105120
- Spectra of quantum graphs–Special topics 129144
- Quantum chaos on graphs 157172
- Some applications and generalizations 181196
- Appendix A. Some notions of graph theory 209224
- Appendix B. Linear operators and operator-functions 213228
- Appendix C. Structure of spectra 219234
- Appendix D. Symplectic geometry and extension theory 223238
- Bibliography 227242
- Index 267282 free
- Back Cover Back Cover1290