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Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

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Abstract

We prove that certain natural random variables associated with the local eigenvalue statistics for generalized lattice Anderson models constructed with finite-rank perturbations are compound Poisson distributed. This distribution is characterized by the fact that the Lévy measure is supported on at most a finite set of positive integers determined by the rank. The proof relies on a Minami-type estimate for finite-rank perturbations. For Anderson-type continuum models on \({{\mathbb{R}^d}}\), we prove a similar result for certain natural random variables associated with the local eigenvalue statistics. We prove that the compound Poisson distribution associated with these random variables has a Lévy measure whose support is at most the set of positive integers.

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Authors and Affiliations

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

    Peter D. Hislop

  2. Institute for Mathematical Sciences, IV Cross Road, CIT Campus, Taramani, Chennai, Tamil Nadu, 600 113, India

    M. Krishna

Authors
  1. Peter D. Hislop
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  2. M. Krishna
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Correspondence to Peter D. Hislop.

Additional information

Communicated by L. Erdös

PDH was partially supported by the NSF through Grant DMS-1103104. MK was partially supported by IMSc Project 12-R&D-IMS-5.01-0106. PDH thanks the IMSc, and MK thanks the Mathematics Department UK, for warm hospitality. The authors thank N. Minami, F. Klopp, D. Dolai, and A. Mallick for discussions on eigenvalue statistics, and the referees for useful remarks.

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Hislop, P.D., Krishna, M. Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations. Commun. Math. Phys. 340, 125–143 (2015). https://doi.org/10.1007/s00220-015-2426-5

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  • DOI: https://doi.org/10.1007/s00220-015-2426-5

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