Spectral Theory of Ergodic Operators

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This research project studies the theory of quantum mechanical phenomena in disordered environments. The research aims to extend mathematical analysis of the electronic properties of disordered structures within the framework of ergodic Schrödinger operators. The project has potential impact in physics through improvement of the understanding of quantum mechanical transport properties of media exhibiting certain kinds of disorder. The project investigates spectral properties of ergodic Schrödinger operators. The potentials of these Schrödinger operators are obtained by sampling with a continuous function along the orbits of an ergodic transformation on a compact metric space. This framework covers many examples of interest, such as almost-periodic potentials and random potentials. The primary objectives of the project are to investigate the following: direct and inverse spectral theory for quasi-periodic Schrödinger operators in one dimension with applications to the Korteweg-de Vries equation, the presence of absolutely continuous spectrum for quasi-periodic Schrödinger operators in higher dimensions, the relationship between decay of gap lengths and smoothness of the potential, the almost periodicity of the Jacobi parameters associated with balanced measures on dynamically defined Cantor sets, the scope of general operator renormalization equations and their applications, one-dimensional self-similar potentials beyond the reach of hyperbolic dynamics, connections between bound states and essential spectrum for perturbations of periodic Schrödinger operators, and the interface between direct and inverse spectral theory for almost periodic Jacobi matrices.