Spectral Theory of Ergodic Operators

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Quantum mechanics is a fundamental branch of physics whose foundations were established during the first half of the twentieth century. The study of quantum mechanical phenomena in disordered environments has been an area of ongoing active study since the 1950's. The mathematical study of electronic properties of disordered structures is carried out within the framework of ergodic Schrodinger operators. This project therefore has a potential impact on physics as it improves our understanding of quantum mechanical transport properties of media exhibiting certain kinds of disorder. Through the training of graduate students the project has in addition an impact on human resource development. The project will study direct and inverse spectral theory for quasi-periodic Schrodinger operators with applications to the KdV equation, the structure of the spectrum and the type of the density of states measure of the square Fibonacci Hamiltonian at intermediate coupling, transport in quantum systems and mechanisms that determine the associated transport exponents, the absolute of continuity of the density of states measure of the weakly coupled Bernoulli-Anderson model in one dimension, connections between bound states and essential spectrum for perturbations of periodic Schrodinger operators, and the interface between direct and inverse spectral theory for almost periodic Jacobi matrices. Methods from spectral theory and dynamical systems will be used in pursuing these goals.