LOCALIZATION AT WEAK DISORDER: SOME ELEMENTARY BOUNDS
Abstract
An elementary proof is given of localization for linear operators H = Ho + λV, with Ho 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 (Ho). 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.
Dedicated to E. H. Lieb, in celebration of his sixtieth birthday