The numerical spectrum of a one-dimensional Schrödinger operator with two competing periodic potentials

We are concerned with the numerical study of a simple one-dimensional Schr\"odinger operator $-\frac 1 2 \Dxx + \alpha q(x)$ with $\alpha \in \Re$, $q(x)=\cos(x)+\eps \cos(kx)$, $\eps >0$ and $k$ being irrational. This governs the quantum wave function of an independent electron within a crystalline lattice perturbed by some impurities whose dissemination induces long-range order only, which is rendered by means of the quasi-periodic potential $q$. We study numerically what happens for various values of $k$ and $\eps$; it turns out that for $k > 1$ and $\eps\ll 1$, that is to say, in case more than one impurity shows up inside an elementary cell of the original lattice, ``impurity bands" appear and seem to be $k$-periodic. When $\eps$ grows bigger than one, the opposite case occurs.