Discrete Schrodinger Operators and Related Models

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This project concerns the spectral theory of Schrodinger operators, a central topic in quantum mechanics. Schrodinger operators describe the movement of an electron in a medium subject to a disordered system. The development of the theory of Schrodinger operators is expected to enhance the understanding of many types of physical phenomena, including conductance, quantum Hall effect, quasicrystals, and graphene. The PI intends to develop new tools to provide rigorous mathematical explanations for these phenomena. The tools that will be developed will also find applications in other branches of mathematics, including harmonic analysis, probability and number theory.This project consists of several parts. One is to study quantum graphs in magnetic fields. The PI intends to understand the topological structure of the spectrum and spectral decompositions. The second part involves studying Laplacians on discrete periodic graphs with a goal of finding the connection between the presence of spectral gaps and the geometric structure of the underlying lattice. The third goal concerns discrete quasi-periodic Schrodinger operators, focusing on several well-known problems including measure of the spectrum, continuity of the spectrum, structure of eigenfunctions in the localization regime and quantum dynamics in the singular continuous regime for quasi-periodic operators. Another goal is to understand Schrodinger operators with potentials generated by skew-shift. These operators, although being completely deterministic, are expected to resemble random features.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.