LOCALIZATION AT WEAK DISORDER - SOME ELEMENTARY BOUNDS

An elementary proof is given of localization for linear operators H = H(o) + lambdaV, with H(o) translation invariant, or periodic, and V(.) a random potential, in energy regimes which for weak disorder (lambda --> 0) are close to the unperturbed spectrum sigma(H(o)). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [4]; the localization regimes discussed in the two works being supplementary. Included also are some general auxiliary results enhancing the method, which now yields uniform exponential decay for the matrix elements [0\P([a,b]) exp(-itH)\x] of the spectrally filtered unitary time evolution operators, with [a,b] in the relevant range.