Spectral Properties of Ergodic Schroedinger Operators

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Spectral Properties of Ergodic Schroedinger OperatorsAbstract of Proposed ResearchSvetlana Jitomirskaya This project is to study spectra and localization type effects for ergodic Schroedinger operators and the study of anomalous absolutely continuous spectrum and anomalous quantum transport in the quasiperiodic and Anderson model-type settings. It is also planned to study several models related to Bloch electrons in constant and/or random magnetic fields. The project involves the continuing development of non-perturbative methods both for the proofs of localization and other related properties as well as for the study of absolutely continuous spectrum. Other important objectives are the study of issues related to Cantor spectra of quasiperiodic operators, it's occurrence, prevalence, and scaling properties, and the development of smooth (rather than analytic) methods.The proposed research investigates the anomalous spectral and diffusive properties of quasiperiodic and other deterministic and random structures. This is basic 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, and quantum chaos theory. The development of the rigorous theory is expected to contribute to the understanding of all three 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.