The objects of study are Schrödinger operators whose potentials 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 spectral properties of such operators are closely linked to the dynamical behavior of an energy-indexed family of cocycles over the given ergodic transformation. The problems the proposer intends to investigate include the following: an analysis of Schrödinger cocycles over uniformly hyperbolic transformations and an analog of the Kotani Support Theorem in this context, sufficient conditions for the absence of non-uniform hyperbolicity for general cocycles over subshifts, eigenvalue statistics for the critical almost Mathieu operator, the structure of the spectrum of square Fibonacci Hamiltonian at intermediate coupling, denseness of uniform hyperbolicity for cocyles over strictly ergodic transformations beyond the continuous category, inverse spectral theory in the presence of absolutely continuous spectrum, direct and inverse spectral theory for quasi-periodic Schrödinger operators with applications to the KdV equation, and the study of measures on the unit circle associated with almost periodic Verblunsky coefficients.
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 Schrödinger 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.
