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Hardcover ISBN:  9780821892114 
Product Code:  SURV/186 
List Price:  $129.00 
MAA Member Price:  $116.10 
AMS Member Price:  $103.20 
eBook ISBN:  9780821894552 
Product Code:  SURV/186.E 
List Price:  $125.00 
MAA Member Price:  $112.50 
AMS Member Price:  $100.00 
Hardcover ISBN:  9780821892114 
eBook ISBN:  9780821894552 
Product Code:  SURV/186.B 
List Price:  $254.00 $191.50 
MAA Member Price:  $228.60 $172.35 
AMS Member Price:  $203.20 $153.20 

Book DetailsMathematical Surveys and MonographsVolume: 186; 2013; 270 ppMSC: Primary 34; 35; 81; 58; 05; 82; 47
A “quantum graph” is a graph considered as a onedimensional 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 quasionedimensional (e.g., “meso” or “nanoscale”) 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, nanosciences, superconductivity theory, etc.
Quantum graphs present many nontrivial 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.
ReadershipGraduate students and research mathematicians interested in this new area of mathematics and its applications.

Table of Contents

Chapters

1. Operators on graphs. Quantum graphs

2. Quantum graph operators. Special topics

3. Spectra of quantum graphs

4. Spectra of periodic graphs

5. Spectra of quantum graphs. Special topics

6. Quantum chaos on graphs

7. Some applications and generalizations

Appendix A. Some notions of graph theory

Appendix B. Linear operators and operatorfunctions

Appendix C. Structure of spectra

Appendix D. Symplectic geometry and extension theory


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A “quantum graph” is a graph considered as a onedimensional 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 quasionedimensional (e.g., “meso” or “nanoscale”) 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, nanosciences, superconductivity theory, etc.
Quantum graphs present many nontrivial 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.
Graduate students and research mathematicians interested in this new area of mathematics and its applications.

Chapters

1. Operators on graphs. Quantum graphs

2. Quantum graph operators. Special topics

3. Spectra of quantum graphs

4. Spectra of periodic graphs

5. Spectra of quantum graphs. Special topics

6. Quantum chaos on graphs

7. Some applications and generalizations

Appendix A. Some notions of graph theory

Appendix B. Linear operators and operatorfunctions

Appendix C. Structure of spectra

Appendix D. Symplectic geometry and extension theory