Introduction to quantum graphs
QA3 .A4 V.186
- Unknown QA3 .A4 V.186
- Kuchment, Peter, 1949-
- Includes bibliographical references (pages 227-266) and index.
- 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 operator-functions
- Appendix C : structure of spectra
- Appendix D : symplectic geometry and extension theory.
- Publisher's Summary
- 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.
(source: Nielsen Book Data)
- 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.
- Publication date
- Copyright date
- Gregory Berkolaiko, Peter Kuchment.
- Mathematical surveys and monographs ; volume 186