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

We are concerned with the numerical study of a simple one-dimensional Schrodinger operator -1/2 partial derivative(xx) + alpha q(x) with alpha is an element of R, q(x) = cos(x) + epsilon cos(kx), epsilon > 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 epsilon; it turns out that for k > 1 and epsilon << 1, that is to say, in case more than one impurity shows lip inside an elementary cell of the original lattice, "impurity bands" appear and seem to be k-periodic. When e grows bigger than one, the opposite case occurs.