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On the Local Eigenvalue Spacings for Certain Anderson-Bernoulli Hamiltonians

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

Abstract

The aim of this work is to extend the results from Bourgain (On eigenvalue spacings for the 1D Anderson model with singular site distribution) on local eigenvalue spacings to certain 1D lattice Schrodinger with a Bernoulli potential. We assume the disorder satisfies a certain algebraic condition that enables one to invoke the recent results from Bourgain (An application of group expansion to the Anderson-Bernoulli model. Preprint) on the regularity of the density of states. In particular we establish Poisson local eigenvalue statistics in those models.

Keywords

  • Local Eigenvalue Statistics
  • Eigenvalue Spacing
  • Bernoulli Potential
  • Anderson-Bernoulli Model
  • Bourgain

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References

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

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

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  3. J. Bourgain, On eigenvalue spacings for the 1D Anderson model with singular site distribution, in Geometric Aspects of Functional Analysis: Israel Seminar (GAFA) 2011–2013, ed. by B. Klartag, E. Milman (Springer, Heidelberg, 2014, in this volume)

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

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

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Acknowledgements

The author is grateful to the UC Berkeley mathematics department (where the paper was written) for its hospitality. This work was partially supported by NSF grant DMS-1301619.

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Correspondence to Jean Bourgain .

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© 2014 Springer International Publishing Switzerland

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Bourgain, J. (2014). On the Local Eigenvalue Spacings for Certain Anderson-Bernoulli Hamiltonians. 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_7

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