Spectral Theory of Schrodinger Operators and Localization Type Effects in Disordered Environments

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Proposal: DMS-9706443 PI: Svetlana Jitomirskaya This research involves localization type effects for Schrodinger operators and for quantum spin systems, percolation, and contact processes in disordered environments, and the general study of spectra and wave functions of Schrodinger operators. The main objective is to develop methods of proving localization type effects for Schrodinger operators with deterministic potentials. The other goal is to study the relation between spectral and quantum-dynamical properties and long time behavior as they relate to generalized eigenfunctions, particularly in the multidimensional case. Another goal of the proposed research is to study singular continuous spectrum that exhibits critical behavior, particularly for models where it appears for critical values (or intervals) of the parameters. The proposed research is centered around the fundamental properties of disordered systems that serve as models of systems with impurities. Impurities can substantially change various properties of the media and can thus be used to create materials with desired properties. In just this way transistors are made of materials with impurities. Disordered systems are also used in modeling many micro and macro effects: from quantum localization to earthquakes. The proposed topics include studying properties of highly disordered systems of Quantum Mechanics and of systems at critical levels of disorder that demonstrate highly unstable behavior.