Uniform Spectral Properties of One-Dimensional Quasicrystals, III. α-Continuity

We study the spectral properties of one-dimensional whole-line Schrödinger operators, especially those with Sturmian potentials. Building upon the Jitomirskaya–Last extension of the Gilbert–Pearson theory of subordinacy, we demonstrate how to establish α-continuity of a whole-line operator from power-law bounds on the solutions on a half-line. However, we require that these bounds hold uniformly in the boundary condition. We are able to prove these bounds for Sturmian potentials with rotation numbers of bounded density and arbitrary coupling constant. From this we establish purely α-continuous spectrum uniformly for all phases. Our analysis also permits us to prove that the point spectrum is empty for all Sturmian potentials.