Some Abstract Wegner Estimates with Applications

Sabri, Mostafa

doi:10.1007/s11005-013-0666-x

Cited by 3 publications

(7 citation statements)

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“…As was already said, Sabri improved his EVC bounds in [20]. It is worth noticing that this procedure preserves the lower edge of the values of the random variables , which may be convenient in the Lifshitz-type analysis of the bandedge localization.…”

Section: Optimal Evc Bounds For Multiparticle Systemsmentioning

confidence: 84%

“…Recently, Sabri [20] improved the results of [14,16] and obtained optimal one-and two-volume EVC bounds for the multiparticle systems. As usual, special efforts are required in his proof (building on [32]) to treat singular (continuous) marginal distributions.…”

Section: Optimal Evc Bounds For Multiparticle Systemsmentioning

confidence: 96%

“…Sabri [20] proposed a simple proof, building on the works by Stollmann [32] and by Boutet de Monvel et al (cf. [21]) and making use of the so-called uncertainty principles for spectral projections.…”

mentioning

confidence: 99%

“…His method applies both to single-and multiparticle operators; in the latter case, both the one-volume and two-volume estimates were obtained, in a simple way, with the optimal volume dependence. We note that the approach of [20] applies to operators in discrete and continuous configuration spaces, including countable graphs, Euclidean spaces, and the metric (a.k.a. quantum) graphs.…”

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confidence: 99%

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Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

Chulaevsky

2013

International Journal of Statistical Mechanics

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Section: Optimal Evc Bounds For Multiparticle Systemsmentioning

confidence: 84%

Section: Optimal Evc Bounds For Multiparticle Systemsmentioning

confidence: 96%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

Chulaevsky

2013

International Journal of Statistical Mechanics

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“…Hence the E j ( · , ω J c ) : R J → R satisfy the hypotheses of Stollmann's lemma (see [42] and [38]) for any ω J c , so we get…”

Section: Wegner Estimatesmentioning

confidence: 95%

Anderson Localization for a Multi-Particle Quantum Graph

Sabri

2014

Rev. Math. Phys.

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We study a multi-particle quantum graph with random potential. Taking the approach of multiscale analysis we prove exponential and strong dynamical localization of any order in the Hilbert-Schmidt norm near the spectral edge. Apart from the results on multi-particle systems, we also prove Lifshitz-type asymptotics for singleparticle systems. This shows in particular that localization for single-particle quantum graphs holds under a weaker assumption on the random potential than previously known.

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Universality of smoothness of Density of States in arbitrary higher-dimensional disorder under non-local interactions I. From Viéte--Euler identity to Anderson localization

Chulaevsky

2016

Preprint

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It is shown that in a large class of disordered systems with non-degenerate disorder, in presence of non-local interactions, the Integrated Density of States (IDS) is at least Hölder continuous in one dimension and universally infinitely differentiable in higher dimensions. This result applies also to the IDS in any finite volume subject to the random potential induced by an ambient, infinitely extended disordered media. Dimension one is critical: in the Bernoulli case, within the class of exponential interactions, the IDS measure undergoes continuity phase transitions, from absolutely continuous to singular continuous behaviour (the singularity in the latter case was known before). The continuity transitions do not occur for sub-exponential or slower decaying interactions, nor for d ≥ 2. Technically, the case of polynomial decay is the simplest one.The proposed approach provides a complement to the classical Wegner estimate which says, essentially, that the IDS in the short-range models is at least as regular as the marginal distribution of the disorder. In the models with non-local interaction the IDS is actually much more regular than the underlying disorder, which can even be discrete, due to the smoothing effect of multiple convolutions. In turn, smoothness of the IDS is responsable for a mechanism complementing the usual Lifshitz tails phenomenon.It is also shown that the disorder can take various forms (e.g., substitution or random displacements) and need not be stochastically stationary (as in Delone-Anderson or trimmed/crooked Hamiltonians, for example); this does not affect the main phenomena observed already in the simplest setting.Contrary to the situation with the usual lattice Bernoulli-Anderson Hamiltonians, the proof of Anderson localization in the models with infinite-range interaction follows in a fairly standard way from the main bounds on the finite-volume IDS; taking into account a considerable size of the current text, the proof of localization is now presented in a separate work. Another distinction from the approach developed by Bourgain and Kenig for the continuous Bernoulli-Anderson Hamiltonians, and later extended by Germinet and Klein to arbitrary (locally IID) disorder, is that all nontrivial marginal distributions are treated in a unified way, via harmonic analysis and without reduction to an embedded Bernoulli model, thus keeping potential benefits from less singular forms of underlying disorder.Long-range models have an amazingly large number of connections to several classical problems of harmonic analysis, probability theory, dynamical systems and number theory.

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Some Abstract Wegner Estimates with Applications

Cited by 3 publications

References 44 publications

Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

Stochastic Regularization and Eigenvalue Concentration Bounds for Singular Ensembles of Random Operators

Anderson Localization for a Multi-Particle Quantum Graph

Universality of smoothness of Density of States in arbitrary higher-dimensional disorder under non-local interactions I. From Viéte--Euler identity to Anderson localization

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