Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Peter D. Hislop; M. Krishna

doi:10.1007/s00220-015-2426-5

Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Peter D. Hislop, M. Krishna

Mathematics

Research output: Contribution to journal › Article › peer-review

12 Scopus citations

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 $${{\mathbb{R}^d}}$$^Rd, 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.

Original language	English
Pages (from-to)	125-143
Number of pages	19
Journal	Communications in Mathematical Physics
Volume	340
Issue number	1
DOIs	https://doi.org/10.1007/s00220-015-2426-5
State	Published - Nov 14 2015

Bibliographical note

Publisher Copyright:
© 2015, Springer-Verlag Berlin Heidelberg.

Funding

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.

Funders	Funder number
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China
U.S. Department of Energy Chinese Academy of Sciences Guangzhou Municipal Science and Technology Project Oak Ridge National Laboratory Extreme Science and Engineering Discovery Environment National Science Foundation National Energy Research Scientific Computing Center National Natural Science Foundation of China	1103104
Institute of Mathematical Sciences India	D-IMS-5.01-0106

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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Cite this

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title = "Eigenvalue Statistics for Random Schr{\"o}dinger Operators with Non Rank One Perturbations",

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 L{\'e}vy 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 \$\$\{\{\textbackslash{}mathbb\{R\}\textasciicircum{}d\}\}\$\$Rd, 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 L{\'e}vy measure whose support is at most the set of positive integers.",

author = "Hislop, \{Peter D.\} and M. Krishna",

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year = "2015",

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AU - Krishna, M.

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N2 - 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 $${{\mathbb{R}^d}}$$Rd, 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.

AB - 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 $${{\mathbb{R}^d}}$$Rd, 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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Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Abstract

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