Localization at Weak Disorder: Some Elementary Bounds

An elementary proof is given of localization for linear operators H = H o + λV, with H o translation invariant, or periodic, and V (·) a random potential, in energy regimes which for weak disorder (λ → 0) are close to the unperturbed spectrum σ (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.