Hamiltonian Systems and Related Phenomena

Texas A&M University, College Station TX

Investigators

Abstract

This project will study the dynamics of the Schrodinger operator that describes the wave function or state function of a quantum mechanical system. The principal investigator will develop fundamental tools to understand many types of phenomena in physics and chemistry, such as the quantum Hall effect, spin, crystals and quasicrystals. In particular, the project focuses on the conductance, transport, and localization-diffusion in both quasiperiodic and disordered media. The development of the rigorous theory is expected to have many applications, including to quantum information theory and semiconducting materials. The project provides research training opportunities for undergraduate and graduate students, and supports related outreach activities. This project aims at studying several topics in areas spanning mathematical physics, dynamical systems, harmonic analysis, geometric analysis, and algebraic geometry. The Principal Investigator will focus on transitions, hierarchical structures of eigenfunctions, quantum dynamics and spectral gaps for one dimensional quasiperiodic operators and further explore multifrequency and higher dimensional cases. The research will combine methods from algebraic geometry with tools from analysis to study the (ir)reducibility of Bloch, Fermi, and Floquet varieties arising from periodic elliptic operators. In the area of geometric analysis, the project emphasizes using piecewise constructions, gluing techniques, and the compact perturbation theory to investigate the transition of singular spectra and the generic phenomena of Laplacians on noncompact complete manifolds. 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.