Spectral Properties of a Class of Reflectionless Schr\"odinger Operators
  We prove that one-dimensional reflectionless Schr\"odinger 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 Schr\"odinger operators.   In addition, we construct examples of reflectionless Schr\"odinger operators with more general types of spectra, given by the complement of a Denjoy-Widom-type domain, which exhibit a singular component. 
Fritz Gesztesy and Peter Yuditskii