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Decorrelation estimates for random Schr\"odinger operators with non rank one perturbations

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Peter D. Hislop, M. Krishna

Mathematical Physics · Submitted 20 May 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, associated with two energies $E$ and $E'$ satisfying $|E - E'| > 4d$, are independent.

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arXiv ID1505.05218 (version 1)
Pith IDBEMUZPIW, Pith's permanent record of this paper and its review
CitationM. Krishna, Peter D. Hislop. "Decorrelation estimates for random Schr\"odinger operators with non rank one perturbations." arXiv:1505.05218 (2015). https://pith.science/paper/BEMUZPIW
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arXiv listing abstract v1. Results and the main claim are underlined; .

We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, associated with two energies $E$ and $E'$ satisfying $|E - E'| > 4d$, are independent. That is, if $I,J$ are two bounded intervals, the random variables $\xi^\omega_{E}(I)$ and $\xi^\omega_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose L\'evy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. contribution 0.99 0.95 0.24 asserted 1.00
s1 These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, associated with two energies $E$ and $E'$ satisfying $|E - E'| > 4d$, are independent. result 0.93 0.93 0.50 asserted 1.00
s2 That is, if $I,J$ are two bounded intervals, the random variables $\xi^\omega_{E}(I)$ and $\xi^\omega_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose L\'evy measure has finite support. result 0.94 0.86 0.27 asserted 1.00
s3 We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation. result 0.94 0.89 0.14 asserted 1.00

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The paper
Original abstract

We prove decorrelation estimates for generalized lattice Anderson models on $Z^d$ constructed with finite-rank perturbations in the spirit of Klopp \cite{klopp}. These are applied to prove that the local eigenvalue statistics $\xi^\omega_{E}$ and $\xi^\omega_{E^\prime}$, associated with two energies $E$ and $E'$ satisfying $|E - E'| > 4d$, are independent. That is, if $I,J$ are two bounded intervals, the random variables $\xi^\omega_{E}(I)$ and $\xi^\omega_{E'}(J)$, are independent and distributed according to a compound Poisson distribution whose L\'evy measure has finite support. We also prove that the extended Minami estimate implies that the eigenvalues in the localization region have multiplicity at most the rank of the perturbation.

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  • The extended Minami estimate [ref] implies that we need to estimate: { (Tr E_H_ω,ℓ (I_L (E)) ≤m_k ) ∩(Tr E_H_ω,ℓ (J_L(E^')) ≤m_k ) } In fact, we consider the more general estimate: { (Tr E_H_ω,L (I_L(E)) = k_1 ) ∩(Tr E_H_ω,L (J_L(E^')) = k_2 ) }, where $k_1, k_2$ are positive integers independent of $L$.

    Eigenvalue statistics for random Schrodinger operators with non rank one perturbations · Asymptotic independence and decorrelation estimates · 0.82

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