Spectral Theory and Quantum Dynamics

William Marsh Rice University, Houston TX

Investigators

Abstract

This project aims to improve 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 graduate student training, the supervision of undergraduate research, and the writing and publication of an introductory textbook on ordinary differential equations aimed at introducing undergraduate students to a rigorous treatment of this field. This project addresses the spectral analysis of Schrödinger operators and unitary analogues of Jacobi operators. These operators are relevant in many areas, primarily in quantum mechanics and approximation theory. The objective is to develop new approaches for the spectral analysis of these operators, and the methods employed range from functional analysis via harmonic analysis to dynamical systems and ergodic theory. The project also investigates the Schrödinger time evolution in terms of transport and dispersion phenomena. The investigator seeks a complete spectral analysis of Schrödinger operators with potentials generated by hyperbolic transformations, a proof of several conjectures in the context of orthogonal polynomials on the unit circle, an approach to proving zero-measure spectrum for multi-frequency Schrödinger operators, and a study of quantum dynamics from the perspective of transport exponents and dispersive estimates. 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.