Multiplicity bound of singular spectrum for higher rank Anderson models

Mallick, Anish

doi:10.1016/j.jfa.2017.02.018

Cited by 6 publications

(10 citation statements)

References 32 publications

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Section: Bounds On Eigenvalue Multiplicitysupporting

confidence: 56%

“…Further investigations on the simplicity of eigenvalues for Anderson-type models may be found in the article by Naboko, Nichols, and Stolz [17], Mallick [12], Mallick and Krishna [13], and Mallick and Narayanan [14]. Mallick [12] proves that the singular spectrum is simple for a class of Anderson models with higher rank perturbation extending the results of Naboko, Nichols, and Stolz [17]. Mallick and Krishna [13] prove that, for higher rank Anderson models with the single site potential having support in the whole real line, the Minami estimate implies simplicity of the pure point spectrum away from the continuous spectrum.…”

Section: Bounds On Eigenvalue Multiplicitymentioning

confidence: 99%

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Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Hislop

Krishna

2015

Commun. Math. Phys.

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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 determined by the rank. The proof relies on a Minami-type estimate for finite-rank perturbations. For Anderson-type continuum models on 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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Section: Bounds On Eigenvalue Multiplicitysupporting

confidence: 56%

Section: Bounds On Eigenvalue Multiplicitymentioning

confidence: 99%

Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Hislop

Krishna

2015

Commun. Math. Phys.

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“…It should be noted that our result (Theorem 1.1) extends the work of Jakšić and Last [15,17], Naboko, Nichols, and Stolz [27], and Mallick [26] in the following way. In the case of Jakšić and Last [15,17], since the rank of each operator P n is one, the theorem above establishes the simplicity of the singular spectrum.…”

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confidence: 79%

A theorem on the multiplicity of the singular spectrum of a general Anderson-type Hamiltonian

Dolai

Mallick

2021

J. Spectr. Theory

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In this work, we study the multiplicity of the singular spectrum for operators of the form A ! D A C P n ! n C n on a separable Hilbert space H, where A is a self-adjoint operator and ¹C n º n is a countable collection of non-negative finite-rank operators. When ¹! n º n are independent real random variables with absolutely continuous distributions, we show that the multiplicity of the singular spectrum is almost surely bounded above by the maximum algebraic multiplicity of the eigenvalues of the operator p C n .A ! z/ 1 p C n for all n and almost all .z; !/. The result is optimal in the sense that there are operators for which the bound is achieved. We also provide an effective bound on the multiplicity of the singular spectrum for some special cases.

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“…For a comprehensive study of the subject we refer to any of [18,19,20,21,22] and the references there. The next set of questions concern the simplicity of the spectrum and in this direction there are several papers starting from Simon [23], Jakšić-Last [24,25], Naboko-Nichols-Stolz [26], Mallick [27,28,29,30,31]. From these set of papers, we now know that when the rank of P n is one or for some special cases of higher rank P n , the singular spectrum is simple.…”

Section: Introductionmentioning

confidence: 99%

Global multiplicity bounds and spectral statistics for random operators

Mallick

Krishna

2020

Rev. Math. Phys.

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In this paper, we consider Anderson type operators on a separable Hilbert space where the random perturbations are finite rank and the random variables have full support on R. We show that spectral multiplicity has a uniform lower bound whenever the lower bound is given on a set of positive Lebesgue measure on the point spectrum away from the continuous one. We also show a deep connection between the multiplicity of pure point spectrum and local spectral statistics, in particular we show that spectral multiplicity higher than one always gives non-Poisson local statistics in the framework of Minami theory.In particular for higher rank Anderson models with pure-point spectrum, with the randomness having support equal to R, there is a uniform lower bound on spectral multiplicity and in case this is larger than one the local statistics is not Poisson.

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Multiplicity bound of singular spectrum for higher rank Anderson models

Cited by 6 publications

References 32 publications

Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

A theorem on the multiplicity of the singular spectrum of a general Anderson-type Hamiltonian

Global multiplicity bounds and spectral statistics for random operators

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