Cantor Singular Continuous Spectrum for Operators Along Interval Exchange Transformations

TL;DR

This paper demonstrates that Schrödinger operators associated with interval exchange transformations typically have a Cantor spectrum of measure zero and exhibit pure singular continuous spectral type for almost all points.

Contribution

It establishes the spectral properties of Schrödinger operators along interval exchange transformations for the first time.

Findings

Cantor spectrum of measure zero for these operators

Pure singular continuous spectrum for almost all points

Results hold for Lebesgue almost every interval exchange transformation

Abstract

It is shown that Schroedinger operators, with potentials along the shift embedding of Lebesgue almost every interval exchange transformations, have Cantor spectrum of measure zero and pure singular continuous for Lebesgue almost all points of the interval.