Norm estimates of almost Mathieu operators

Abstract

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