On the spectrum of a non-self-adjoint quasiperiodic operator

We study the operator $\mathscr{A}$ acting in $l^2(\mathbb{Z})$ by the formula $(\mathscr{A}u)_l=u_{l+1}+u_{l-1}+\lambda e^{-2\pi i(\theta+\omega l)}u_l$. Here, $l$ is an integer variable, while $\lambda>0$, $\theta\in[0,1)$, and $\omega\in(0,1)$ are parameters. For $\omega\notin\mathbb{Q}$, this is the simplest non-self-adjoint quasiperiodic operator. By means of a renormalization technique, we describe the geometry of the spectrum of this operator, compute the Lyapunov exponent on the spectrum, and describe the conditions under which either the spectrum is pure continuous or a point spectrum appears additionally.