From Uncertainty Principles to Wegner Estimates

Stollmann, Peter

doi:10.1007/s11040-010-9072-0

Cited by 9 publications

(20 citation statements)

References 13 publications

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“…This definition immediately implies that an absolutely continuous measure µ with bounded density ρ, satisfies s(µ, ǫ) ≤ ρ ∞ ǫ. Therefore the Theorem 3.3 of Stollman [54], implies that s(ν, ǫ) ≤ 6 B φ s(µ, ǫ) ≤ C ρ ∞ ǫ. This inequality implies that the density ofν is bounded.…”

Section: )mentioning

confidence: 86%

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Regularity of the Density of States of Random Schrödinger Operators

Dolai

Krishna

Mallick

2020

Commun. Math. Phys.

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Section: )mentioning

confidence: 86%

“…We give the Lemma below which is a consequence of proofs of results in Stollmann [54] and Combes-Hislop-Klopp [13]. These papers essentially prove the result, but we write it here since it does not occur in the form we need to use.…”

Section: Acknowledgementmentioning

confidence: 99%

Regularity of the Density of States of Random Schrödinger Operators

Dolai

Krishna

Mallick

2020

Commun. Math. Phys.

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“…In fact, in the case of H 0 = − this leads to the uncertainty principles considered in harmonic analysis (see e.g. [8] and discussion in [15]). …”

Section: An Uncertainty Principle and Mobility Of The Ground State Enmentioning

confidence: 99%

An uncertainty principle, Wegner estimates and localization near fluctuation boundaries

2010

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We prove a simple uncertainty principle and show that it can be applied to prove Wegner estimates near fluctuation boundaries. This gives new classes of models for which localization at low energies can be proven. IntroductionStarting point of the present paper was the lamentable fact that for certain random models with possibly quite small and irregular support there was a proof of localization via fractional moment techniques (at least for d ≤ 3) but no proof of Wegner estimates necessary for multiscale analysis. The classes of models include models with surface type random potentials as well as Anderson models with displacement (see [1]) but actually much more classes of examples could be seen in the framework established there which was labelled "fluctuation boundaries". Actually, the big issue in the treatment of random perturbations with small or irregular support is the question, whether the spectrum at low energies really feels the random perturbation. This is exactly what is formalized in the fluctuation boundary framework.In the present paper we establish the necessary Wegner estimates by using the method from Combes et al. [6] so that we get the correct volume factor and the modulus of continuity of the random variables. One of the main ideas we borrow from the last mentioned work is A. Boutet

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“…The proof had been given by Combes, Hislop and Klopp in [1], proof of Thm 1.1 and we stress the fact that in this part of their work no periodicity assumptions are used. The main difference to the claim in [2] is that we loose the very explicit control of the constant; apart from that all the results remain valid.…”

Section: The Error and What Remains Truementioning

confidence: 82%