Spectral Homogeneity of Limit-Periodic Schrödinger Operators

Abstract

We prove that the spectrum of a limit-periodic Schrödinger 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 Schrödinger 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.

To appear in Journal of Spectral Theory