Ergodic Schrödinger Operators

This project aims to improve the understanding of how the amount of disorder present in an environment can promote or suppress transport in a system. This issue is studied in the context of quantum mechanics at the atomic level. Applications of new insights about quantum systems include the development of quantum computing devices and quantum algorithms. The project supports education and diversity though the mentoring of postdoctoral scholars, the training of graduate students, and the supervision of undergraduate research. This project addresses the general theory of Schrödinger operators with ergodic potentials. These operators are relevant in many areas, primarily in quantum mechanics and approximation theory. The objective is to establish results for general base transformations and for large classes of sampling functions. The methods employed range from functional analysis via harmonic analysis to dynamical systems and ergodic theory. The investigator seeks to identify the almost sure spectral type of an ergodic family of Schrödinger operators, while establishing a version of Simon's Wonderland Theorem in this setting and answering a question of Walters about the existence of non-uniform cocycles as byproducts, to develop further gap labelling theory based on the Schwartzman group, along with a comparison with gap labelling based on K-theory, to study the Laplacian on Penrose and other aperiodically ordered tilings, and to obtain proofs of Cantor spectra via cocycle perturbation techniques beyond the two-dimensional time-discrete setting. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.