Uniform spectral properties of one-dimensional quasicrystals, III. $α$-continuity

Abstract

We study the spectral properties of discrete one-dimensional Schrödinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely $α$-continuous spectrum, uniformly for all phases. The proofs rely on the unique decomposition property of Sturmian potentials, a mass-reproduction technique based upon a Gordon-type argument, and on the Jitomirskaya-Last extension of the Gilbert-Pearson theory of subordinacy.

12 pages