Positivity of the Lyapunov exponent for analytic quasiperiodic operators with arbitrary finite-valued background

TL;DR

This paper establishes that for certain quasiperiodic Schrödinger operators with finite-valued backgrounds, the Lyapunov exponent remains positively bounded from below at large coupling, indicating strong localization properties.

Contribution

It provides a uniform lower bound on the Lyapunov exponent for a broad class of quasiperiodic operators with finite-valued backgrounds at high coupling.

Findings

Lyapunov exponent is positive for large coupling constants.

The lower bound on the Lyapunov exponent is uniform across energies and backgrounds.

Results imply strong localization in the studied operators.

Abstract

We study lower bounds on the Lyapunov exponent associated with one-frequency quasiperiodic Schr\"odinger operators with an added finite valued background potential. We prove that, for sufficiently large coupling constant, the Lyapunov exponent is positive with a uniform (in energy and background) minoration.