Norm estimates of almost Mathieu operators

TL;DR

This paper provides improved estimates for the norm of the almost Mathieu operator within the rotation C*-algebra, refining previous bounds and contributing to the understanding of its spectral properties.

Contribution

The authors significantly improve the existing inequality for the norm of the almost Mathieu operator in a specific parameter range.

Findings

Enhanced upper bound for the operator norm

Refined inequality for $ heta \

Abstract

We estimate the norm of the almost Mathieu operator $H_{\theta,\lambda} =U+U^*+(\lambda /2)(V+V^*)$ in the rotation $C^*$-algebra $A_\theta =C^*(U,V unitaries;UV=e^{2\pi i\theta} VU)$. In this process, we significantly improve the inequality $\| H_\theta \| \leq 2\sqrt{2}$, $\theta \in [0.25,0.5]$, conjectured by Beguin, Valette and Zuk.