Uniform spectral properties of one-dimensional quasicrystals, I. Absence of eigenvalues

TL;DR

This paper proves that for a broad class of one-dimensional quasicrystals modeled by Sturmian potentials, there are no eigenvalues, indicating purely continuous spectral behavior for most cases.

Contribution

It establishes the absence of eigenvalues for a full-measure set of rotation numbers in Sturmian potentials, including the Fibonacci case, advancing understanding of spectral properties.

Findings

Absence of eigenvalues for Sturmian potentials with full-measure rotation numbers

Results apply to all elements in the hull, indicating uniform spectral properties

Includes the significant Fibonacci case among the full-measure set

Abstract

We consider discrete one-dimensional Schr\"odinger operators with Sturmian potentials. For a full-measure set of rotation numbers including the Fibonacci case we prove absence of eigenvalues for all elements in the hull.