## Localization at Weak Disorder: Some Elementary Bounds (1993)

Citations: | 75 - 6 self |

### BibTeX

@MISC{Aizenman93localizationat,

author = {Michael Aizenman},

title = {Localization at Weak Disorder: Some Elementary Bounds},

year = {1993}

}

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### OpenURL

### Abstract

An elementary proof is given of localization for linear operators H=H o +lV, with H o translation invariant, or periodic, and V( . ) a random potential, in energy regimes which for weak disorder (l®0) are close to the unperturbed spectrum s(H o ). The analysis is within the approach introduced in the recent study of localization at high disorder by Aizenman and Molchanov [AM]; 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] e -itH |x> of the spectrally filtered unitary time evolution operators, with [a,b] in the relevant energy range. corrected 7/12/93 Localization at Weak Disorder 2 1. Introduction This work presents an elementary derivation of localization for time evolutions generated by linear operators consisting of a translation invariant, or periodic, part and an added random potential, at energy rang...