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

We study the spectral properties of one-dimensional whole-line Schrodinger operators, especially those with Sturmian potentials. Building upon the Jit omirskaya-Last extension of the Gilbert-Pearson theory of subordinacy, we d emonstrate how to establish alpha- 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 num bers of bounded density and arbitrary coupling constant. From this we estab lish purely alpha-continuous spectrum uniformly for all phases. Our analysis also permits us to prove that the point spectrum is empty for all Sturmian potentials.