LEAPS-MPS: Ergodic Jacobi Matrices

The study of quantum systems is fundamental in modern mathematical physics. This project aims to study the long-time behavior of quantum systems by building new bridges between direct and inverse approaches. The direct approach asks one to describe the attributes of a given system, while the inverse approach asks what systems may exhibit specified attributes.

The project plans to support education and diversity though a summer school in mathematical physics, the supervision of undergraduate research, and the writing and publication of a graduate textbook on ergodic Schrödinger operators aimed at introducing graduate students to this field.

This project addresses the spectral analysis of Schrödinger, Jacobi, and Dirac operators with coefficients obtained by continuously sampling along the orbits of an ergodic topological dynamical system. This framework includes many models of interest, such as crystals, quasicrystals, and disordered media. The project will study open questions related to the quantum evolution for such models as well as the structure of the spectrum for pseudo-random models, periodic operators on graphs, and aperiodic tilings.

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.