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On Eigenvalue Spacings for the 1-D Anderson Model with Singular Site Distribution

Part of the Lecture Notes in Mathematics book series (LNM,volume 2116)

Abstract

We study eigenvalue spacings and local eigenvalue statistics for 1D lattice Schrödinger operators with Hölder regular potential, obtaining a version of Minami’s inequality and Poisson statistics for the local eigenvalue spacings. The main additional new input are regularity properties of the Furstenberg measures and the density of states obtained in some of the author’s earlier work.

Keywords

  • Eigenvalue Spacing
  • Site Distribution
  • Anderson Localization
  • Wegner Estimate
  • Anderson-type Models

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Ph. Bougerol, J. Lacroix, Products of Random Matrices with Applications to Schrödinger Operators (Birkhauser, Boston, 1985)

    CrossRef  MATH  Google Scholar 

  2. J. Bourgain, On the Furstenberg measure and density of states for the Anderson-Bernoulli model at small disorder. J. Anal. Math. 117, 273–295 (2012)

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  3. J. Bourgain, An application of group expansion to the Anderson-Bernoulli model. Preprint 07/13

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  4. J. Bourgain, On the local eigenvalue spacings for certain Anderson-Bernoulli Hamiltonians. Preprint 08/13

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  5. R. Carmona, A. Klein, G. Martinelli, Anderson localization for Bernoulli and other singular potentials. Commun. Math. Phys. 108, 41–66 (1987)

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  6. J.-M. Combes, F. Germinet, A. Klein, Generalized eigenvalue - counting estimates for the Anderson model. J. Stat. Phys. 135, 201–216 (2009)

    CrossRef  MathSciNet  MATH  Google Scholar 

  7. F. Germinet, F. Klopp, Spectral statistics for random Schrödinger operators in the localized regime. JEMS (to appear)

    Google Scholar 

  8. C. Shubin, T. Vakilian, T. Wolff, Some harmonic analysis questions suggested by Anderson-Bernoulli models. Geom. Funct. Anal. 8, 932–964 (1988)

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Acknowledgements

The author is grateful to an anonymous referee and A. Klein for comments and to the UC Berkeley mathematics department for their hospitality. This work was partially supported by NSF grant DMS-1301619.

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Correspondence to Jean Bourgain .

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Bourgain, J. (2014). On Eigenvalue Spacings for the 1-D Anderson Model with Singular Site Distribution. In: Klartag, B., Milman, E. (eds) Geometric Aspects of Functional Analysis. Lecture Notes in Mathematics, vol 2116. Springer, Cham. https://doi.org/10.1007/978-3-319-09477-9_6

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