Norm estimates of discrete Schrödinger operators

Harper's operator is defined on $\ell^2({\sym Z})$ by $$ H_\theta \xi(n) = \xi(n+1) + \xi(n-1) + 2\cos n\theta\, \xi(n), $$ where $\theta\! \in \![0,\pi]$. We show that the norm of $\|H_\theta\|$ is less than or equal to $2\sqrt{2}$ for $\pi/2 \le\theta\le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.