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## and localization near fluctuation boundaries (2009)

### Citations

114 |
Bootstrap multiscale analysis and localization in random media
- Germinet, Klein
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Citation Context ...ugh and η ∈ L∞ with supp η ⊂[E0, E0 + δ] it follows that E {∥ ∥ p |X| η (H(ω)) · χK ∥ } < ∞ for every compact K ⊂ R d . Remark (1) Maybe one can strengthen the estimate of Theorem 3.1 in the sense of =-=[7]-=-. Note, however, that in the latter paper a stronger Wegner estimate is supposed to hold. (2) The theorem provides an extension to d ≥ 4 of the main result of [1]. Moreover, there is no technique at t... |

97 | The uncertainty principle in harmonic analysis. Ergebnisse der Mathematik und ihrer Grenzgebiete
- Havin, Jöricke
- 1994
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Citation Context ...nty principle as a state in the range of PI cannot be “concentrated where W vanishes”. In fact, in the case of H0 =−�this leads to the uncertainty principles considered in harmonic analysis (see e.g. =-=[8]-=- and discussion in [15]). In our main application, H0 will be a Schrödinger operator so that (∗) is in fact a variant of the usual uncertainty principle. The use of (∗) for the proof of Wegner estimat... |

62 |
Bounds on the density of states of disordered systems.
- Wegner
- 1981
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Citation Context ...ring in the conditional probabilities are not displayed correctly. For a detailed discussion of regular conditional probabilities see, e.g., [12]. Wegner estimates, named after Wegner’s original work =-=[17]-=-, are an important tool in random operator theory. 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, se... |

59 | An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators, preprint 2006
- Combes, Hislop, et al.
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Citation Context ... 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 de Monvel (B) Institut de Mat... |

44 |
Existence and regularity properties of the integrated density of states of random Schrödinger
- Veselić
- 2007
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Citation Context ...m variables in the Wegner estimate as well as the correct volume factor. The models we consider needn’t have a homogeneous background so that the integrated density of states, IDS need not exist. See =-=[16]-=- for a recent survey on how to prove the existence of the IDS in various different settings. We show that a straightforward application of Theorem 1.1 above gives the necessary input to perform the an... |

41 | Hölder continuity of the integrated density of states for some random operators at all energies
- Combes, Hislop, et al.
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Citation Context ... main application, H0 will be a Schrödinger operator so that (∗) is in fact a variant of the usual uncertainty principle. The use of (∗) for the proof of Wegner estimates is due to Combes et al., see =-=[5,6]-=-. Its importance lies in the fact that it takes care of random potentials with small support. Our purpose here is to prove a simple criterion that implies (∗) and can be checked rather easily. Theorem... |

37 | Wegner estimates and Anderson localization for alloy-type potentials - Kirsch - 1996 |

30 | Wegner bounds for a two-particle tight binding model - Chulaevsky, Suhov |

17 | Veselic: Bounds on the spectral shift function and the density of states - Hundertmark, Killip, et al. - 2005 |

15 | Localization near fluctuation boundaries via fractional moments and applications
- Monvel, Naboko, et al.
(Show Context)
Citation Context ...r 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 perturbati... |

14 |
Probability theory. Universitext. Springer-Verlag London Ltd
- Klenke
- 2008
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Citation Context ...3) also appears in [6], where, however, the variables appearing in the conditional probabilities are not displayed correctly. For a detailed discussion of regular conditional probabilities see, e.g., =-=[12]-=-. Wegner estimates, named after Wegner’s original work [17], are an important tool in random operator theory. They give a bound on the probability that the eigenvalues of a local Hamiltonian come clos... |

12 | Wegner estimate for sparse and other generalized alloy type potentials - Kirsch, Veselić |

10 |
Caught by disorder, volume 20
- Stollmann
- 2001
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Citation Context ...rably extends the main result of [1]. (3) At the same time, the estimates that come out of our analysis are weaker than those in the latter paper. Sketch of the proof We use the multiscale setup from =-=[14]-=-. By now it is quite well understood that homogeneity does not play a major role so that multiscale analysis goes through without 123An uncertainty principle, Wegner estimates and localization near f... |

5 | Fron Uncertainty Principles to Wegner Estimates
- Stollmann
- 2011
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Citation Context ...te in the range of PI cannot be “concentrated where W vanishes”. In fact, in the case of H0 =−�this leads to the uncertainty principles considered in harmonic analysis (see e.g. [8] and discussion in =-=[15]-=-). In our main application, H0 will be a Schrödinger operator so that (∗) is in fact a variant of the usual uncertainty principle. The use of (∗) for the proof of Wegner estimates is due to Combes et ... |

4 |
A Wegner-type estimate for correlated potentials
- Chulaevsky
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Citation Context ...e above result is that the important constant is controlled in a simple way. (2) Once the ground state energy is pushed up by W we get an uncertainty principle (∗) at least for intervals I near λ(0). =-=(3)-=- The corresponding uncertainty result in [5] for periodic Schrödinger operators does not follow from the preceding theorem. There is a kind of converse to Theorem 1.1. Lemma 1.2 If (∗) holds for I wit... |

4 | Continuity of integrated density of states—independent randomness - Krishna - 2007 |

3 |
Wegner-Stollmann type estimates for some quantum lattice systems
- Chulaevsky
- 2007
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Citation Context ...ily implies validity of (∗)forκ= supt>0 t statement. . This is the desired ⊓⊔ Remark (1) One particularly nice aspect of the above result is that the important constant is controlled in a simple way. =-=(2)-=- Once the ground state energy is pushed up by W we get an uncertainty principle (∗) at least for intervals I near λ(0). (3) The corresponding uncertainty result in [5] for periodic Schrödinger operato... |