Spectral theory of ergodic Schrodinger operators and related models

University Of California-Irvine, Irvine CA

Investigators

Abstract

The proposed research concerns the study of the fundamental properties of disordered systems. Disordered systems are models of systems with impurities. The development of a rigorous theory of disordered systems is expected to contribute to the understanding of many types of physical phenomena, and in particular, may lead to finding new materials with desired physical properties. Disordered systems are also used in modeling many other micro and macro effects from quantum localization- an important topic in quantum computation- to earthquakes. An integral part of the project concerns educating graduate students and other young researchers. The PI will also continue her outreach efforts to entice K-12 students to the study of mathematics. The project consists of several parts. One is to prove the extended states for multidimensional quasiperiodic operators. Another is to study the effects of interaction in tight-binding quasi-periodic models. One more is to study local eigenvalue statistics in the regime of localization for discrete ergodic Schroedinger operators. It is also planned to study several models related to Bloch electrons in constant magnetic fields, notably the Extended Harper Model. Other important objectives are the study of issues related to Cantor/non-Cantor spectra of quasiperiodic operators. The project involves the continuing development of non-perturbative methods for the proofs of localization type effects both in Schrodinger operators and in quantum spin systems, percolation and contact processes in disordered environments, as well as for the study of absolutely continuous spectrum.