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Correlation Estimates in the Anderson Model

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Abstract

We give a new proof of correlation estimates for arbitrary moments of the resolvent of random Schrödinger operators on the lattice that generalizes and extends the correlation estimate of Minami for the second moment. We apply this moment bound to obtain a new n-level Wegner-type estimate that measures eigenvalue correlations through an upper bound on the probability that a local Hamiltonian has at least n eigenvalues in a given energy interval. Another consequence of the correlation estimates is that the results on the Poisson statistics of energy level spacing and the simplicity of the eigenvalues in the strong localization regime hold for a wide class of translation-invariant, selfadjoint, lattice operators with decaying off-diagonal terms and random potentials.

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Authors and Affiliations

  1. Georgia Institute of Technology, School of Mathematics, Atlanta, GA, 30332-0160, USA

    Jean V. Bellissard

  2. Department of Mathematics, University of Kentucky, Lexington, KY, 40506-0027, USA

    Peter D. Hislop

  3. Department of Mathematics, UAB, Birmingham, AL, 35294-1170, USA

    Günter Stolz

Authors
  1. Jean V. Bellissard
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  2. Peter D. Hislop
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  3. Günter Stolz
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Corresponding author

Correspondence to Peter D. Hislop.

Additional information

Authors supported in part by NSF grants DMS 06009565 (J.V.B.), 0503784 (P.D.H.), and 0245210 (G.S.).

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Bellissard, J.V., Hislop, P.D. & Stolz, G. Correlation Estimates in the Anderson Model. J Stat Phys 129, 649–662 (2007). https://doi.org/10.1007/s10955-007-9409-7

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  • DOI: https://doi.org/10.1007/s10955-007-9409-7

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