Hausdorff Dimension of Spectrum of One-Dimensional Schrödinger Operator with Sturmian Potentials

Abstract

Let β∈(0,1) be an irrational, and [a1,a2,...] be the continued fraction expansion of β. Let Hβ be the one-dimensional Schrodinger operator with Sturmian potentials. We show that if the potential strength V>20, then the Hausdorff dimension of the spectrum σ(Hβ) is strictly great than zero for any irrational β, and is strictly less than 1 if and only if lim inf k→∞(a1a2⋅⋅⋅ak))1/k<∞.