Norm estimates of almost Mathieu operators

Abstract

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

18 pages,3 figures