Duke Mathematical Journal

Lifshitz tails and localization in the three-dimensional Anderson model

Alexander Elgart

Abstract

Consider the three-dimensional Anderson model with a zero mean and bounded independent, identically distributed random potential. Let $\lambda$ be the coupling constant measuring the strength of the disorder, and let $\sigma(E)$ be the self-energy of the model at energy $E$. For any $\epsilon{>}0$ and sufficiently small $\lambda$, we derive almost-sure localization in the band $E\le -\sigma(0)-\lambda^{4-\epsilon}$. In this energy region, we show that the typical correlation length $\xi_E$ behaves roughly as $O\big((|E|-\sigma(E))^{-1/2}\big)$, completing the argument outlined in the preprint of T. Spencer [18]

Article information

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

Dates
First available in Project Euclid: 5 January 2009

https://projecteuclid.org/euclid.dmj/1231170943

Digital Object Identifier
doi:10.1215/00127094-2008-068

Mathematical Reviews number (MathSciNet)
MR2477764

Zentralblatt MATH identifier
1165.82015

Citation

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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