Unique Continuation Principle for Spectral Projections of Schrödinger Operators and Optimal Wegner Estimates for Non-ergodic Random Schrödinger Operators

Abstract

We prove a unique continuation principle for spectral projections of Schrödinger operators. We consider a Schrödinger operator H = −∆ + V on L 2 (R d ), and let H Λ denote its restriction to a finite box Λ with either Dirichlet or periodic boundary condition. We prove unique continuation estimates of the typewith κ > 0 for appropriate potentials W ≥ 0 and intervals I. As an application, we obtain optimal Wegner estimates at all energies for a class of non-ergodic random Schrödinger operators with alloytype random potentials ('crooked' Anderson Hamiltonians). We also prove optimal Wegner estimates at the bottom of the spectrum with the expected dependence on the disorder (the Wegner estimate improves as the disorder increases), a new result even for the usual (ergodic) Anderson Hamiltonian. These estimates are applied to prove localization at high disorder for Anderson Hamiltonians in a fixed interval at the bottom of the spectrum.

Wegner Estimate and Disorder Dependence for Alloy-Type Hamiltonians with Bounded Magnetic Potential

“…We prove an optimal Wegner estimate valid at all energies. The proof is an adaptation of arguments from [16], combined with a recent quantitative unique continuation estimate for eigenfunctions of elliptic operators from [1]. This generalizes Klein's result to operators with a bounded magnetic vector potential.…”

“…However, so far one was not able to prove such a Wegner estimate for alloy-type models with and without magnetic field above the threshold E(∞), cf. [26,4,16].…”

“…Let us give a brief overview here. We refer to [16] and [30] for further references. In [5], an optimal Wegner estimate at all energies is proved for the usual (ergodic) Anderson Hamiltonian under the additional assumption that a covering condition holds.…”

“…In the non-ergodic setting, optimal Wegner estimates have been proved, e.g., in [21] for the Landau Hamiltonian, and up to a logarithmic correction in [22] for non-magnetic Schrödinger operators. The logarithmic factor in the energy is removed in [16].…”

“…Let us define the class of random Schrödinger operators studied in this note. It is a generalization of the models studied in [16]. The non-random part is given by the selfadjoint magnetic Schrödinger operator…”