Overview

This guide presents a comprehensive treatment of the mathematical physics of quantum wires and devices. The focus is on results in the area of the spectral theory of bent and deformed quantum wires, simple quantum devices, Anderson localization, the quantum Hall effect and graphical models for quantum wire systems. The Selberg trace formula for finite volume graphical models is reviewed. Examples and relationships to advances in acoustic and fluid flow, trapped states and spectral resonances, quantum chaos, random matrix theory, spectral statistics, point interactions, photonic crystals, Landau models, quantum transistors, edge states and metal-insulator transitions are developed. Problems related to modelling open quantum devices are discussed. The research of Exner and co-workers in quantum wires, Stollmann, Figotin, Bellissard et al in the area of Anderson localization and the quantum Hall effect, and Bird, Ferry, Berggren and others in the area of quantum devices and their modelling is surveyed. The work on finite volume graphical models is interconnected to recent work on Ramanujan graphs and diagrams, the Phillips-Sarnak conjectures, L-functions and scattering theory. This text should be of use to physicists, mathematicians and engineers interested in quantum wires, quantum devices and related mesoscopic systems.

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