Spectral theory of ergodic Schrodinger operators and related models

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The project consists of two main parts. One is to study the effects of interaction in tight-binding quasi-periodic models. The other is to study local eigenvalue statistics in the regime of localization for discrete ergodic Schrodinger operators. A related project is to prove or disprove singularity of the integrated density of states for the Anderson-Bernoulli model. It is also planned to study several models related to Bloch electrons in constant or random magnetic fields. 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.The proposed research concerns the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is therefore research on the fundamental properties of disordered systems that serve as models of systems with impurities. Quasiperiodic operators provide central or important models for integer quantum Hall effect, experimental quasicrystals, quantum chaos theory, and the theory of graphene. The development of the rigorous theory is expected to contribute to the understanding of all the above 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 to earthquakes. The proposed topics include studying properties of both highly and weakly disordered systems of Quantum Mechanics that demonstrate certain anomalous behavior. An integral part of the project concerns educating graduate students. It is also planned to continue the related outreach activities.