Almost Mathieu operator

Example

Let $\lambda, \alpha, \theta$ be real numbers. The almost Mathieu operator $H_{\lambda,\alpha,\theta}$ acts on the Hilbert space $\ell^2(\Z)$ of square-summable sequences $u = (u(n))_{n \in \Z}$ by $$(H_{\lambda,\alpha,\theta}u)(n) = u(n + 1) + u(n - 1) + 2\lambda\cos\big(2\pi(n\alpha + \theta)\big)u(n).$$ It is a bounded self-adjoint operator. Check: each of the three terms has norm at most $1$, $1$ and $2\abs{\lambda}$, so $\norm{H_{\lambda,\alpha,\theta}} \le 2 + 2\abs{\lambda}$. The shifts $u(n) \mapsto u(n + 1)$ and $u(n) \mapsto u(n - 1)$ are adjoint to each other, so their sum is self-adjoint, and multiplication by a real sequence is self-adjoint.