By Berkolaiko G., Kuchment P.
A "quantum graph" is a graph regarded as a one-dimensional complicated and outfitted with a differential operator ("Hamiltonian"). Quantum graphs come up certainly as simplified versions in arithmetic, physics, chemistry, and engineering while one considers propagation of waves of assorted nature via a quasi-one-dimensional (e.g., "meso-" or "nano-scale") method that appears like a skinny local of a graph. Works that at the moment will be labeled as discussing quantum graphs were showing due to the fact that at the least the Thirties, and because then, quantum graphs thoughts were utilized effectively in numerous parts of mathematical physics, arithmetic quite often and its purposes. you'll point out, for example, dynamical structures conception, regulate concept, quantum chaos, Anderson localization, microelectronics, photonic crystals, actual chemistry, nano-sciences, superconductivity thought, and so forth. Quantum graphs current many non-trivial mathematical demanding situations, which makes them expensive to a mathematician's middle. paintings on quantum graphs has introduced jointly instruments and instinct coming from graph thought, combinatorics, mathematical physics, PDEs, and spectral conception. This e-book presents a entire creation to the subject, gathering the most notions and strategies. It additionally encompasses a survey of the present nation of the quantum graph examine and purposes