Absolutely continuous spectrum for limit-periodic Schr\"odinger operators

TL;DR

This paper proves that many limit-periodic Schrödinger operators in arbitrary dimensions have purely absolutely continuous spectra, extending known results from one dimension and employing a non-perturbative construction of extended states.

Contribution

It generalizes the absolute continuity of the spectrum for limit-periodic Schrödinger operators to higher dimensions using novel probabilistic estimates.

Findings

Purely absolutely continuous spectrum in arbitrary dimensions

Non-perturbative construction of extended states

New estimate on eigenvalue simplicity probability

Abstract

We show that a large class of limit-periodic Schr\"odinger operators has purely absolutely continuous spectrum in arbitrary dimensions. This result was previously known only in dimension one. The proof proceeds through the non-perturbative construction of limit-periodic extended states. An essential step is a new estimate of the probability (in quasi-momentum) that the Floquet Bloch operators have only simple eigenvalues.