Abstract. In this paper we investigate numerically the spectrum of some representative
examples of discrete one-dimensional Schr¨odinger operators with
quasi-periodic potential in terms of a perturbative constant b and the spectral
parameter a. Our examples include the well-known Almost Mathieu model,
other trigonometric potentials with a single quasi-periodic frequency and generalisations
with two and three frequencies. We computed numerically the
rotation number and the Lyapunov exponent to detect open and collapsed
gaps, resonance tongues and the measure of the spectrum.
We found that the case with one frequency was 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 we found that there is a curve in the (a, b)-plane which separates
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.