Limit-Periodic Schr\"odinger Operators in the Regime of Positive Lyapunov Exponents

Abstract

We investigate the spectral properties of the discrete one-dimensional Schr\"odinger operators whose potentials are generated by continuous sampling along the orbits of a minimal translation of a Cantor group. We show that for given Cantor group and minimal translation, there is a dense set of continuous sampling functions such that the spectrum of the associated operators has zero Hausdorff dimension and all spectral measures are purely singular continuous. The associated Lyapunov exponent is a continuous strictly positive function of the energy. It is possible to include a coupling constant in the model and these results then hold for every non-zero value of the coupling constant.