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

TL;DR

This paper investigates the spectral characteristics of one-dimensional Schrödinger operators with Sturmian potentials, demonstrating the absence of point spectrum and establishing purely α-continuous spectrum for certain rotation numbers.

Contribution

It proves the absence of point spectrum and the presence of purely α-continuous spectrum for Sturmian potentials with bounded density rotation numbers, extending spectral theory results.

Findings

Point spectrum is always empty.

Purely α-continuous spectrum established for bounded density rotation numbers.

Results are uniform across all phases.

Abstract

We study the spectral properties of discrete one-dimensional Schr\"odinger operators with Sturmian potentials. It is shown that the point spectrum is always empty. Moreover, for rotation numbers with bounded density, we establish purely $\alpha$-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.