Bounds on the Spectral Shift Function and the Density of States

Hundertmark, Dirk; Killip, Rowan; Nakamura, Shu; Stollmann, Peter; Veselić, Ivan

doi:10.1007/s00220-005-1460-0

Cited by 33 publications

(45 citation statements)

References 53 publications

(74 reference statements)

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“…In previous papers [6,7], we proved global Hölder continuity, for any order strictly less than one, of the IDS under the same hypotheses on the single-site probability measure, and, in [18], there was an improvement up to a logarithmic factor (see below). It has long been expected that if the probability measure of a single-site random variable has a bounded density with compact support, then the IDS should be locally Lipschitz continuous at all energies.…”

Section: Introduction and Main Resultsmentioning

confidence: 77%

“…In Theorem 1.3, this covering condition is no longer necessary. The result in [18] follows from a new exponentially decreasing bound, in the index n, on the n th singular value of the difference of two semigroups generated by Hamiltonians H 1 and H 2 for which the perturbation H 1 − H 2 has compact support. This estimate is used to improve the estimate on the spectral shift function obtained in [10].…”

Section: Theorem 13 Assume That the Family Of Random Schrödinger Opementioning

confidence: 97%

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An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

2007

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We prove that the integrated density of states (IDS) of random Schrödinger operators with Anderson-type potentials on L 2 (R d ), for d ≥ 1, is locally Hölder continuous at all energies with the same Hölder exponent 0 < α ≤ 1 as the conditional probability measure for the single-site random variable. As a special case, we prove that if the probability distribution is absolutely continuous with respect to Lebesgue measure with a bounded density, then the IDS is Lipschitz continuous at all energies. The single-site potential u ∈ L ∞ 0 (R d ) must be nonnegative and compactly-supported. The unperturbed Hamiltonian must be periodic and satisfy a unique continuation principle. We also prove analogous continuity results for the IDS of random Anderson-type perturbations of the Landau Hamiltonian in two-dimensions. All of these results follow from a new Wegner estimate for local random Hamiltonians with rather general probability measures.

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Section: Introduction and Main Resultsmentioning

confidence: 77%

Section: Theorem 13 Assume That the Family Of Random Schrödinger Opementioning

confidence: 97%

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

2007

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“…Using Lemma 6 in [7] we have [φ n (ω e(n) + 4ε) − φ n (ω e(n) )] dµ(ω e(n) ) ≤ s(µ, 4ε)[φ n (b + 4ε) − φ n (a)]. Let supp(µ) ⊂ (a, b).…”

Section: Proofsmentioning

confidence: 98%

The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs

Gruber¹,

Veselić²

2008

Random Operators and Stochastic Equations

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We consider an alloy type potential on an infinite metric graph. We assume a covering condition on the single site potentials. For random Schrödingers operator associated with the alloy type potential restricted to finite volume subgraphs we prove a Wegner estimate which reproduces the modulus of continuity of the single site distribution measure. The Wegner constant is independent of the energy.

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“…They give a bound on the probability that the eigenvalues of a local Hamiltonian come close to a given energy. For a list of some (recent) papers, see [2][3][4][5][6][9][10][11]13] and the account in the recent Lecture Notes Volume [16]. We consider a box ⊂ R d and denote by H (ω) the restriction of H (ω) to L 2 ( ) with Dirichlet boundary conditions and with H 0 the restriction of H 0 to L 2 ( ) with Dirichlet boundary conditions.…”

Section: Continuity Of the Ids Near Weak Fluctuation Boundariesmentioning

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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Bounds on the Spectral Shift Function and the Density of States

Cited by 33 publications

References 53 publications

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators

The modulus of continuity of Wegner estimates for random Schrödinger operators on metric graphs

An uncertainty principle, Wegner estimates and localization near fluctuation boundaries

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