Introduction to quantum graphs
 Author/Creator
 Berkolaiko, Gregory, 1976
 Language
 English.
 Publication
 Providence, Rhode Island : American Mathematical Society, [2013]
 Copyright notice
 ©2013
 Physical description
 xiii, 270 pages : illustrations ; 27 cm.
 Series
 Mathematical surveys and monographs ; no. 186.
Access
Available online

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QA3 .A4 V.186
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Contributors
 Contributor
 Kuchment, Peter, 1949
Contents/Summary
 Bibliography
 Includes bibliographical references (pages 227266) and index.
 Contents

 Operators on graphs : quantum graphs
 Quantum graph operators : special topics
 Spectra of quantum graphs
 Spectra of periodic graphs
 Spectra of quantum graphs : special topics
 Quantum chaos on graphs
 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.
 Publisher's Summary
 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.
(source: Nielsen Book Data)
Subjects
 Subject
 Quantum graphs.
 Boundary value problems.
 Combinatorics  Graph theory  Graphs and linear algebra (matrices, eigenvalues, etc.)
 Ordinary differential equations  Boundary value problems  Boundary value problems on graphs and networks.
 Partial differential equations  Spectral theory and eigenvalue problems  Spectral theory and eigenvalue problems.
 Partial differential equations  Miscellaneous topics  Partial differential equations on graphs and networks (ramified or polygonal spaces)
 Operator theory  Miscellaneous applications of operator theory  Applications in chemistry and life sciences.
 Global analysis, analysis on manifolds  Partial differential equations on manifolds; differential operators  Spectral problems; spectral geometry; scattering theory.
 Quantum theory  General mathematical topics and methods in quantum theory  Quantum mechanics on special spaces: manifolds, fractals, graphs, etc.
 Quantum theory  General mathematical topics and methods in quantum theory  Quantum chaos.
 Statistical mechanics, structure of matter  Applications to specific types of physical systems  Superconductors.
 Statistical mechanics, structure of matter  Applications to specific types of physical systems  Quantum wave guides, quantum wires.
 Statistical mechanics, structure of matter  Applications to specific types of physical systems  Nanostructures and nanoparticles.
Bibliographic information
 Publication date
 2013
 Copyright date
 2013
 Responsibility
 Gregory Berkolaiko, Peter Kuchment.
 Series
 Mathematical surveys and monographs ; volume 186
 ISBN
 9780821892114 (alk. paper)
 0821892118 (alk. paper)