A lower bound on the Lyapunov exponent for the generalized Harper's model

TL;DR

This paper establishes a new lower bound for the Lyapunov exponent of a generalized Harper's model, extending previous bounds by including additional parameters, which enhances understanding of the model's spectral properties.

Contribution

It introduces a novel lower bound for the Lyapunov exponent that accounts for multiple parameters, surpassing the classical Herman's bound.

Findings

Derived a lower bound incorporating both $a_1$ and $a_2$

Extended the understanding of spectral properties of the model

Improved bounds over previous results

Abstract

We obtain a lower bound for the Lyapunov exponent of a family of discrete Schr\"{o}dinger operators $(Hu)_n=u_{n+1}+u_{n-1}+2a_1\cos2\pi(\theta+n\alpha)u_n+2a_2\cos4\pi(\theta+n\alpha)u_n$, that incorporates both $a_1$ and $a_2,$ thus going beyond the Herman's bound.