Spectral Homogeneity of Limit-Periodic Schr\"odinger Operators

TL;DR

This paper demonstrates that limit-periodic Schrödinger operators with potentials satisfying the Pastur--Tkachenko condition have homogeneous spectra, leading to a dense set with purely absolutely continuous spectrum on a homogeneous Cantor set.

Contribution

It establishes spectral homogeneity for a class of limit-periodic Schrödinger operators and links spectral properties to the potential's almost periodicity characteristics.

Findings

Spectra are homogeneous under the Pastur--Tkachenko condition.

A dense set of such operators have purely absolutely continuous spectrum.

Spectral gaps can be infinite if the potential is not uniformly almost periodic.

Abstract

We prove that the spectrum of a limit-periodic Schr\"odinger operator is homogeneous in the sense of Carleson whenever the potential obeys the Pastur--Tkachenko condition. This implies that a dense set of limit-periodic Schr\"odinger operators have purely absolutely continuous spectrum supported on a homogeneous Cantor set. When combined with work of Gesztesy--Yuditskii, this also implies that the spectrum of a Pastur--Tkachenko potential has infinite gap length whenever the potential fails to be uniformly almost periodic.