Mathematical Sciences: Singular Continuous Spectum and Localization Type Effects if Disordered Systems

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9501265 Jitomirskaya This research involves Schrodinger operators with singular continuous spectrum, localization type effects for Sahrodinger operators and for quantum spin systems, percolation and contact processes in disordered environments. The main objective of the research related to the singular continuous spectrum is to study the relation between the spectral properties and longtime behavior of the generalized eigenfunctions, in particular between certain growth properties of generalized eigenfunctions, the Hausdorff dimension of the spectral measure, and stability or instability of singular continuous spectrum with respect to rank one perturbations. Another goal of the proposed research is to study singular continuous spectrum for models where singular spectrum may appear for the critical values (intervals) of the parameters, serving as a transition from absolutely continuous to pure point spectrum. The main objective of the research related to localization is to develop methods of proving localization for ergodic families of operators (disordered systems) with deterministic disorder. Also general uniformity properties of localization and relation to quantum dynamics will be studied. The proposed research is centered around the fundamental properties of disordered and singular systems that are used in modeling of many micro and macro effects: from quantum localization to earthquake theory. The proposed topics include studying properties of highly disordered systems of Quantum ald Statistical Mechanics and of systems at critical levels of disorder, that demonstrate highly unstable behavior. ***