03-377 Michael Aizenman, Alexander Elgart, Sergey Naboko, Jeffrey H. Schenker and Gunter Stolz

Moment Analysis for Localization in Random Schr\"odinger Operators (922K, postscript) Aug 21, 03

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Abstract. We study localization effects of disorder on the spectral and dynamical properties of Schr\"odinger-type operators with random potentials. The new results include a fractional-moment method for continuum operators, which enables us to establish exponentially decaying bounds for the mean values of transition amplitudes, and of related resolvent operator kernels, for energies throughout the localization regime. The obstacles which have up to now prevented an extension to the continuum of this method, initially developed in the discrete context, are traced to the lack of a uniform bound on the Lifshitz-Krein spectral shift associated with the local potential terms, for which in the discrete case there is a simple universal upper bound. The difficulties are resolved here through an analysis of the resonance-diffusing effects of the disorder.

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