Spectral Properties of Limit-Periodic Schr\"odinger Operators

Abstract

We investigate the spectral properties of Schr\"odinger operators in l^2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show that for any such base dynamics, the spectrum is a Cantor set of positive Lebesgue measure and purely absolutely continuous for a dense set of sampling functions, and it is a Cantor set of zero Lebesgue measure and purely singular continuous for a dense G_\delta set of sampling functions.