Resonance tongues and spectral gaps in quasi-periodic Schrödinger operators with one or more frequencies. A numerical exploration (Q650178)

The authors present a numerical exploration on the spectrum of some representative examples of discrete one-dimensional Schrödinger operators with quasi-periodic potential in terms of a perturbative constant \(b\) and the spectral parameter \(a\). Their examples include the well-known almost Mathieu model, other trigonometric potentials with a single quasi-periodic frequency and generalizations with two and three frequencies. The rotation number and the Lyapunov exponent are computed numerically to detect open and collapsed gaps, resonance tongues and the measure of the spectrum. Here, it is observed that the case with one frequency is significantly different from the case of several frequencies because the latter has all gaps collapsed for a sufficiently large value of the perturbative constant and thus the spectrum is a single spectral band with positive Lyapunov exponent. In contrast, in the cases with one frequency considered, gaps are always dense in the spectrum, although some gaps may collapse either for a single value of the perturbative constant or for a range of values. In all cases there is a curve in the \((a,b)\)-plane separating the regions where the Lyapunov exponent is zero in the spectrum and where it is positive. Along this curve, which is \(b=2\) in the almost Mathieu case, the measure of the spectrum is zero.