06-232 V.Grinshpun

Pure Absolutely Continuous Spectrum for Random Operators on l^2(Z^d) at Low Disorder (2054K, PDF) Aug 26, 06

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Abstract. Absence of singular continuous component, with probability one, in the spectra of random perturbations of multidimensional finite-difference Hamiltonians, is for the first time rigorously established under certain conditions ensuring either absence of point component, or absence of absolutely continuous component in the corresponding regions of spectra. The main technical tool involved is the rank-one perturbation theory of singular spectra. The respective new result (the non-mixing property) is applied to establish existence and bounds of the (non-empty) pure absolutely continuous component in the spectrum of the Anderson model with bounded random potential in dimension d=2 at low disorder (similar proof holds for d>4). The new result implies, via the trace-class perturbation analysis, Anderson model with the unbounded random potential having only pure point spectrum (complete system of localized wave-functions) with probability one in arbitrary dimension. The basic idea is to establish absence of the mixed, point and continuous, spectra in the range of the conductivity spectral component of the arbitrary (bounded non-random) perturbation, it had been understood by author (1999) while independent study of the exactly solvable model, and of the disordered surface model (explicitly considered in the paper). Various generalizations are applicable to describe the spectral properties of multidimensional Hamiltonians with Anderson-type potentials, random and non-random as well (subject to the possible forthcoming communication by the author). The new results imply the non-zero value of conductivity in the energy regime corresponding to the high impurity concentration and zero temperature (at low disorder), providing rigorous proof for the so-called Mott conjecture.

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