Spectral Properties of a Class of Reflectionless Schrödinger Operators

Abstract

We prove that one-dimensional reflectionless Schrödinger operators with spectrum a homogeneous set in the sense of Carleson, belonging to the class introduced by Sodin and Yuditskii, have purely absolutely continuous spectra. This class includes all earlier examples of reflectionless almost periodic Schrödinger operators. In addition, we construct examples of reflectionless Schrödinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain, which exhibit a singular component.