Work

On Gaps in the Spectra of Quasiperiodic Schrödinger Operators with Discontinuous Monotone Potentials

Year: 2025

Type: article

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 results 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.

Source: International Mathematics Research Notices

DOI: 10.1093/imrn/rnaf085

Authors: Ilya Kachkovskiy, Leonid Parnovski, Roman Shterenberg

Institutions: Michigan State University, University College London, University of Alabama at Birmingham

Language: English

Topic: Spectral Theory in Mathematical Physics

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,534

Funders: National Science Foundation, Leverhulme Trust, Engineering and Physical Sciences Research Council