Bandwidths for a quasiperiodic tight-binding model

A study is made of the spectrum of a class of one-dimensional models that are equivalent to the equation for an electron in a magnet field and a two-dimensional periodic potential. A rigorous lower bound for the measure of the spectrum is derived, and theoretical and numerical arguments are presented to show that this bound is attained in the incommensurate limit. In the case that corresponds to an isotropic system this lower bound is zero, and numerical work shows that the measure has the asymptotic form $\frac{A}{p}$, where $p$ is the period. The existence of a finite Lyapunov exponent and of a nonzero spectral measure in the incommensurate limit seems to be correlated with the existence of semiclassical open orbits in the problem of an electron in a magnetic field.