Localization for Random Perturbations of Periodic Schrödinger Operators with Regular Floquet Eigenvalues

Abstract

We prove a localization theorem for continuous ergodic Schrödinger operators Hω := H 0 +Vω, where the random potential Vω is a nonnegative Anderson-type perturbation of the periodic operator H 0 . We consider a lower spectral band edge of σ(H 0 ), say E = 0, at a gap which is preserved by the perturbation Vω. Assuming that all Floquet eigenvalues of H 0 , which reach the spectral edge 0 as a minimum, have there a positive definite Hessian, we conclude that there exists an interval I containing 0 such that Hω has only pure point spectrum in I for almost all ω.Since 0 is in the support of the density of ω 0 it follows that 0 ∈ σ(H ω ). In this case we say that 0 is a lower band edge of the periodic operator, which is preserved by the positive random perturbation V ω .

On the Lipschitz continuity of the integrated density of states for sign-indefinite potentials

“…In the energy/disorder regime (i) and the additional assumption that supp f is an interval, the estimate is proven in Proposition 4.2 of [28]. If the support of f has several components we still have Lifshitz tails at the bottom of the spectrum and the statement of Proposition 1.2 in [45] holds with the same proof.…”

“…Without loss of generality we will assume that E 0 is a lower spectral edge such that [30] and [45]). Spectral localization for single site potentials of changing sign in the energy/disorder regimes (i) and (ii) was established in [46].…”

“…Note that, since the random potential of H ω,+ is non-negative, and min supp f = 0, E 0 is a common lower spectral edge of both the unperturbed, periodic operator H 0 and of the random one H ω,+ . Theorem 2.2.1 in [42] proves (5.5) under assumption (ii) and Proposition 1.2 in [45] under assumption (iii). In the energy/disorder regime (i) and the additional assumption that supp f is an interval, the estimate is proven in Proposition 4.2 of [28].…”

The integrated density of states for random Schrödinger operators

“…Like Lifshitz tails at the bottom of the spectrum internal Lifshitz tails can be used as an input for a localization proof [188].…”

2-dimensional localization of acoustic waves in random perturbation of periodic media

“…Strong dynamical localization up to any order and in the Hilbert-Schmidt norm is proved in [13]. See also [2][3][4]6,22,23,39,47]. In these references, localization near the so-called fluctuation boundaries is proved.…”