Spectral Transitions and Critical Phenomena

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The research will focus on the anomalous spectral properties of quasiperiodic (almost but not quite periodic) structures. Quasiperiodic operators provide central or important models for integer quantum Hall effect, experimental quasicrystals, and the quantum chaos theory. Quasiperiodic systems are also used in modeling many other micro and macro effects: from quantum localization to earthquakes. Other deterministic aperiodic (irregular) structures will be of interest as well. The planned development of the rigorous theory is expected to contribute to the understanding of all the above phenomena and may lead to finding new materials with desired physical properties. The topics will include studying properties of quantum mechanical systems with both strong and weak disorder (many and few impurities, respectively), which demonstrate certain anomalous behavior. An integral part of the project will consist of educating graduate students and other young researchers. Related outreach activities will take place.The project consists of several parts, including the connection of dual Lyapunov exponents to characterization of spectra and spectral components, proof of the ubiquity of arithmetic spectral transitions and universal hierarchical structures of eigenfunctions for analytic quasiperiodic operators, proof of extended states for multidimensional quasiperiodic operators, studies of the critical phenomena and of the ‘two interlacing particles’ effect. Other important objectives will be the study of issues related to certain models of quantum chaos. The project will involve the continuing development of non-perturbative methods for the proofs of localization-type effects, as well as for the study of absolutely continuous spectrum.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.