Schr\"odinger Operators Defined by Interval Exchange Transformations

TL;DR

This paper investigates discrete Schr"odinger operators generated by minimal interval exchange transformations, revealing new examples with purely singular spectrum for all non-constant continuous potentials.

Contribution

It provides the first examples of such operators with purely singular spectrum for every non-constant continuous sampling function.

Findings

Operators have purely singular spectrum for all non-constant continuous potentials

First examples of transformations with this spectral property

Spectral and spectral type results for these operators

Abstract

We discuss discrete one-dimensional Schr\"odinger operators whose potentials are generated by an invertible ergodic transformation of a compact metric space and a continuous real-valued sampling function. We pay particular attention to the case where the transformation is a minimal interval exchange transformation. Results about the spectrum and the spectral type of these operators are established. In particular, we provide the first examples of transformations for which the associated Schr\"odinger operators have purely singular spectrum for every non-constant continuous sampling function.