An uncertainty principle, Wegner estimates and localization near fluctuation boundaries

Abstract

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

From Uncertainty Principles to Wegner Estimates

“…The term in square brackets is a structural constant that, typically, does not depend on Λ. As we showed in [3], we can find an uncertainty estimate of the form ( ) with κ independent of Λ for very general Anderson type random models. The remaining terms give the right volume dependence as well as the continuity of the single site random variables.…”

“…In [3] we showed that we have an uncertainty principle ( ) with κ independent of Λ. Moreover, for the models considered there, the other constants are uniformly bounded.…”

“…The above three theorems are mutually incomparable: as shown in [3], Theorem 2.2 applies in situations where H = H(ω) and W may even be concentrated near a subspace of R d , so that W is not "thick" at all; the decisive input is the mobility of the ground state energy. However, we can only treat intervals I close to the ground state energy.…”

“…Subsequently we showed in [3] how to use this approach to deduce localization for a great number of models that couldn't be treated before. Actually, for low energies we derived an uncertainty principle by an extremely simple method.…”

“…However, for the link from these uncertainty principles to Wegner estimates with the correct volume factor we still had to rely upon the analysis of [5] which is technically quite involved. Here, we will use the fact that the uncertainty principles in [3] apply to the spectral projections of the random operators and not just to to the spectral projections of the underlying periodic background, as was the case in [5]. This allows the substantial shortcut we present in this paper.…”