Sharp arithmetic delocalization for quasiperiodic operators with potentials of semi-bounded variation

TL;DR

This paper establishes a sharp arithmetic Gordon's theorem for a broad class of one-dimensional quasiperiodic Schrödinger operators, showing the absence of eigenvalues in certain energy sets without requiring continuity of potentials.

Contribution

It introduces a new uniform upper bound on cocycle iterates of bounded variation, extending spectral analysis to unbounded monotone and bounded variation potentials.

Findings

Absence of eigenvalues for energies with bounded Lyapunov exponent.

Applicable to unbounded monotone potentials and bounded variation potentials.

Provides a sharp arithmetic criterion for spectral properties.

Abstract

We obtain the sharp arithmetic Gordon's theorem: that is, absence of eigenvalues on the set of energies with Lyapunov exponent bounded by the exponential rate of approximation of frequency by the rationals, for a large class of one-dimensional quasiperiodic Schr\"odinger operators, with no (modulus of) continuity required. The class includes all unbounded monotone potentials with finite Lyapunov exponents and all potentials of bounded variation. The main tool is a new uniform upper bound on iterates of cocycles of bounded variation.