Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Hislop, Peter D.; Krishna, M.

doi:10.1007/s00220-015-2426-5

Cited by 15 publications

(16 citation statements)

References 28 publications

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“…Besides the existence of good configurations for clusters of eigenvalues established above, the second ingredient for the proof of Theorem 2.2 is a probabilistic estimate on the maximal size of generic clusters of eigenvalues. For lattice models, such estimates follow from an adaption of the method developed in [15], see [30]. The following assertion extends this idea.…”

Section: Proof Of Theorem 22mentioning

confidence: 79%

“…However, one can still partially carry out this reduction for an arbitrary fixed interval [0, E]. In the discrete setting, this output is sufficient to establish a weaker result, namely compound Poisson statistics, [30]. We expect that an adaptation of the method to our context will show compound Poisson statistics for energies above E sp .…”

Section: Resultsmentioning

confidence: 97%

“…The method from [38] is based on a probabilistic estimate on the event that two or more eigenvalues of H ω,L are located in a small energy window. Such estimates are referred to as Minami estimates and have been further developed in [8,29,15,12,46,30]. However, with the exception of the one-dimensional case [37], these techniques heavily rely on the concrete structure of the random potential V ω in H A ω .…”

Section: Introductionmentioning

confidence: 99%

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Level spacing for continuum random Schrödinger operators with applications

Dietlein¹,

Elgart²

2017

Preprint

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For continuum alloy-type random Schrödinger operators with signdefinite single-site bump functions and absolutely continuous single-site randomness we prove a probabilistic level-spacing estimate at the bottom of the spectrum. More precisely, given a finite-volume restriction of the random operator onto a box of linear size L, we prove that with high probability the eigenvalues below some threshold energy Esp keep a distance of at least e −(log L) β for sufficiently large β > 1. This implies simplicity of the spectrum of the infinite-volume operator below Esp. Under the additional assumption of Lipschitz-continuity of the single-site probability density we also prove a Minami-type estimate and Poisson statistics for the point process given by the unfolded eigenvalues around a reference energy E. Contents 1. Introduction 1 2. Model and results 3 3. Clusters of eigenvalues 9 4. Proof of the level spacing estimates 13 5. Proof of the Minami-type estimate 24 6. Simplicity of spectrum and Poisson statistics 26 Appendix A. Properties of deformed Schrödinger operators 29 Appendix B. Eigenfunction decay for localized energies 30 Acknowledgements 32 References 32

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Section: Proof Of Theorem 22mentioning

confidence: 79%

Section: Resultsmentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Level spacing for continuum random Schrödinger operators with applications

Dietlein¹,

Elgart²

2017

Preprint

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“…) is asymptotically Gaussian for any fixed n. This Gaussian fluctuation result at the bottom of the spectrum is new and should be compared to the conjecture that eigenvalues have local Poisson statistics in spectral regions where localization holds (in particular, at edges of the spectrum other than its bottom) and have random matrix GOE statistics in the bulk of regions where delocalization holds. Rigorous results on Poisson statistics in the localized regime were pioneered by Minami [28] for the Anderson model, and we refer to [17,24,8] and references therein for recent developments, but to our knowledge the problem still remains open for the divergence-form operator −∇ • a∇ apart from the 1D case covered in [31]. Rigorous results on GOE statistics in the delocalized regime are only known in the simplified setting of random band matrix models [7,6].…”

Section: Introductionmentioning

confidence: 99%

Eigenvalue fluctuations for random elliptic operators in homogenization regime

Duerinckx

2021

Preprint

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This work is devoted to the asymptotic behavior of eigenvalues of an elliptic operator with rapidly oscillating random coefficients on a bounded domain with Dirichlet boundary conditions. A sharp convergence rate is obtained for isolated eigenvalues towards eigenvalues of the homogenized problem, as well as a quantitative two-scale expansion result for eigenfunctions. Next, a quantitative central limit theorem is established for eigenvalue fluctuations; more precisely, a pathwise characterization of eigenvalue fluctuations is obtained in terms of the so-called homogenization commutator, in parallel with the recent fluctuation theory for the solution operator.

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“…This is one of the first results on the nature of the limiting eigenvalue point process of the LES for random Schrödinger operators on R d , for d 2. In [12], two of the authors proved that the limit points of the local processes ξ L ω are compound Poisson point processes. In the absence of a Minami-type estimate, this is the best possible result.…”

Section: Introduction: Random Point Interactionsmentioning

confidence: 99%

Eigenvalue statistics for Schrödinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

Hislop,

Kirsch,

Krishna

2019

Preprint

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We prove that the local eigenvalue statistics at energy E in the localization regime for Schrödinger operators with random point interactions on R d , for d = 1, 2, 3, is a Poisson point process with the intensity measure given by the density of states at E times the Lebesgue measure. This is one of the first examples of Poisson eigenvalue statistics for the localization regime of multi-dimensional random Schrödinger operators in the continuum. The special structure of resolvent of Schrödinger operators with point interactions facilitates the proof of the Minami estimate for these models. Dedicated to the memory of Alexandre Grossmann Contents 1. Introduction: Random point interactions 1.1. Contents of the paper 2. Local operators for point interactions with boundary conditions 3. Finite-volume estimates: Wegner, Minami, and localization estimates 3.1. Wegner estimate 3.2. Minami estimate 3.3. Localization estimates 4. Local eigenvalue statistics: independent arrays 4.1. Intensity of the limiting point process 4.2. Elimination of double points 4.3. LES for the uana 5. Approximation of the point process ξ ω by a uana 5.1. Estimation of the first term (5.9) 5.2. Decomposition of the second term (5.10) 5.3. Estimation of the terms involving continuous kernels 6. Conclusion of the proof of Theorem 1.1 on local eigenvalue statistics 7. Appendix: Estimates on rank one perturbations 8. Appendix: Details for dimensions d = 1 and d = 2 References

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Eigenvalue Statistics for Random Schrödinger Operators with Non Rank One Perturbations

Cited by 15 publications

References 28 publications

Level spacing for continuum random Schrödinger operators with applications

Level spacing for continuum random Schrödinger operators with applications

Eigenvalue fluctuations for random elliptic operators in homogenization regime

Eigenvalue statistics for Schrödinger operators with random point interactions on $\mathbb{R}^d$, $d=1,2,3$

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