David Damanik, Rowan Killip, and Daniel Lenz
Uniform spectral properties of one-dimensional quasicrystals, III.
$\alpha$-continuity
(49K, LaTeX)
ABSTRACT. We study the spectral properties of one-dimensional whole-line
Schr\"odinger operators, especially those with Sturmian potentials.
Building upon the Jitomirskaya-Last extension of the Gilbert-Pearson
theory of subordinacy, we demonstrate 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 number of bounded density and arbitrary coupling
constant. From this we establish 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