Norm estimates of discrete Schrödinger operators

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