Half-line eigenfunction estimates and purely singular continuous spectrum of zero Lebesgue measure

Abstract

We consider discrete one-dimensional Schrödinger operators with minimally ergodic, aperiodic potentials taking finitely many values. The well-known tendency of these operators to have purely singular continuous spectrum of zero Lebesgue measures is further elucidated. We provide a unified approach to both the study of the spectral type as well as the measure of the spectrum as a set. Namely, we define a stability set and show that if this set has positive measure, then it implies both absence of eigenvalues almost surely and zero-measure spectrum. As a byproduct we get absence of eigenvalues inside the original spectrum for local perturbations of these operators. We apply this approach to Schrödinger operators with Sturmian potentials. Finally, in the appendix, we discuss the two different strictly ergodic dynamical systems associated to a circle map.