On gaps in the spectra of quasiperiodic Schrödinger operators with discontinuous monotone potentials

Abstract

We show that, for one-dimensional discrete Schrödinger 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.

22 pages. Final accepted journal version

References in corpus (5)

• Dry Ten Martini Problem in the non-critical case
• Kotani theory, Puig's argument, and stability of The Ten Martini Problem
• The Dry Ten Martini Problem for Sturmian Hamiltonians
• Opening Gaps in the Spectrum of Strictly Ergodic Jacobi and CMV Matrices
• Sharp arithmetic localization for quasiperiodic operators with monotone potentials

Cited by in corpus (1)

• Perturbative diagonalization and spectral gaps of quasiperiodic operators on ${\ell }^2({\mathbb{Z}}^d)$ with monotone potentials