Localization for a continuum Cantor-Anderson Hamiltonian

TL;DR

This paper establishes localization at the spectrum's bottom for a continuum Schrödinger operator with a highly singular, Cantor set-supported potential distribution, extending multiscale analysis techniques to this challenging setting.

Contribution

It introduces a novel multiscale analysis approach for continuum operators with singular distributions, generalizing Sperner's Lemma using the LYM inequality and scale-dependent configuration classes.

Findings

Localization at the spectrum's bottom is proven.

Wegner estimate is obtained scale by scale despite singularity.

Method extends multiscale analysis to Cantor set-supported distributions.

Abstract

We prove localization at the bottom of the spectrum for a random Schr\"odinger operator in the continuum with a single-site potential probability distribution supported by a Cantor set of zero Lebesgue measure. This distribution is too singular to be treated by the usual methods. In particular, an "a priori" Wegner estimate is not available. To prove the result we perform a multiscale analysis following the work of Bourgain and Kenig for the Bernoulli-Anderson Hamiltonian, and obtain the required Wegner estimate scale by scale. To do so, we generalize their argument based on Sperner's Lemma by resorting to the LYM inequality for multisets, and combine it with the concept of scale dependent equivalent classes of configurations introduced by Germinet, Hislop and Klein for the study of Poisson Hamiltonians.