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 Schrödinger operator − 1 2 ∂xx + αq(x) with α ∈ R, q(x) = cos(x) + ε cos(kx), 0 ≤ ε ¿ 1 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 ε; especially, increasing values of k correspond to a stronger disorder in the medium and we expect to observe a mobility edge. However, it turns out that for k > 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.