On gaps in the spectra of quasiperiodic Schr\"odinger operators with discontinuous monotone potentials

TL;DR

This paper demonstrates that certain quasiperiodic Schr"odinger operators with discontinuous, monotone potentials have Cantor spectra, by linking spectral gaps to stability of localization under rank one perturbations.

Contribution

It establishes a connection between localization stability and the absence of spectral intervals, proving Cantor spectrum for specific quasiperiodic operators with discontinuous potentials.

Findings

Quasiperiodic operators with sawtooth-like potentials have Cantor spectra.

Stability of Anderson localization implies absence of spectral intervals.

Results on gap filling for non-monotone quasiperiodic operators with discontinuities.

Abstract

We show that, for one-dimensional discrete Schr\"odinger operators, stability of Anderson localization under a class of rank one perturbations implies absence of intervals in spectra. The argument is based on well-known result of Gordon and del Rio--Makarov--Simon, combined with a way to consider perturbations whose ranges are not necessarily cyclic. The main application of the results is showing that a class of quasiperiodic operators with sawtooth-like potentials, for which such a version of stable localization is known, has Cantor spectra. We also obtain several results on gap filling under rank one perturbations for some general (not necessarily monotone) classes of quasiperiodic operators with discontinuous potentials.