Duke Mathematical Journal

Lifshitz tails and localization in the three-dimensional Anderson model

Alexander Elgart

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Consider the three-dimensional Anderson model with a zero mean and bounded independent, identically distributed random potential. Let λ be the coupling constant measuring the strength of the disorder, and let σ(E) be the self-energy of the model at energy E. For any ε>0 and sufficiently small λ, we derive almost-sure localization in the band Eσ(0)λ4ε. In this energy region, we show that the typical correlation length ξE behaves roughly as O((|E|σ(E))1/2), completing the argument outlined in the preprint of T. Spencer [18]

Article information

Duke Math. J., Volume 146, Number 2 (2009), 331-360.

First available in Project Euclid: 5 January 2009

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Zentralblatt MATH identifier

Primary: 82B44: Disordered systems (random Ising models, random Schrödinger operators, etc.) 81T15: Perturbative methods of renormalization
Secondary: 47B80: Random operators [See also 47H40, 60H25] 81Q10: Selfadjoint operator theory in quantum theory, including spectral analysis 81T18: Feynman diagrams


Elgart, Alexander. Lifshitz tails and localization in the three-dimensional Anderson model. Duke Math. J. 146 (2009), no. 2, 331--360. doi:10.1215/00127094-2008-068. https://projecteuclid.org/euclid.dmj/1231170943

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