Spectral Fluctuations for Schrödinger Operators with a Random Decaying Potential

Jonathan Breuer; Yoel Grinshpon; Moshe J. White

doi:10.1007/s00023-021-01082-9

Spectral Fluctuations for Schrödinger Operators with a Random Decaying Potential

Jonathan Breuer, Yoel Grinshpon^*, Moshe J. White

^*Corresponding author for this work

Einstein Institute of Mathematics

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

2 Scopus citations

Abstract

We study fluctuations of polynomial linear statistics for discrete Schrödinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.

Original language	American English
Pages (from-to)	3763-3794
Number of pages	32
Journal	Annales Henri Poincare
Volume	22
Issue number	11
DOIs	https://doi.org/10.1007/s00023-021-01082-9
State	Published - Nov 2021

Bibliographical note

Funding Information:
We are grateful to Ofer Zeitouni for a useful discussion, and to the anonymous referee for useful comments. Research by JB and YG was supported in part by the Israel Science Foundation (Grant No. 399/16) and in part by the United States-Israel Binational Science Foundation (Grant No. 2014337). Research by MW was supported in part by the Israel Science Foundation (Grant No. 1612/17) and in part by the United States-Israel Binational Science Foundation (Grant No. 2006066).

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© 2021, Springer Nature Switzerland AG.

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title = "Spectral Fluctuations for Schr{\"o}dinger Operators with a Random Decaying Potential",

abstract = "We study fluctuations of polynomial linear statistics for discrete Schr{\"o}dinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.",

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N2 - We study fluctuations of polynomial linear statistics for discrete Schrödinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.

AB - We study fluctuations of polynomial linear statistics for discrete Schrödinger operators with a random decaying potential. We describe a decomposition of the space of polynomials into a direct sum of three subspaces determining the growth rate of the variance of the corresponding linear statistic. In particular, each one of these subspaces defines a unique critical value for the decay-rate exponent, above which the random variable has a limit that is sensitive to the underlying distribution and below which the random variable has asymptotically Gaussian fluctuations.

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Spectral Fluctuations for Schrödinger Operators with a Random Decaying Potential

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