Scaling, localization and bandwidths for equations with competing periods

The finite size scaling theory of the measure of the spectrum of Harper's equation is reexamined. For sequences of fractions tending to a rational limit a simple criterion is derived which determines whether the corrections to scaling behave as (log p)−2 or p−2 as the denominator p is increased. The question of how special are the properties of the Harper equation, is studied. It is shown that if the pure cosine term in the diagonal term is replaced by a distorted periodic function the different subbands undergo a transition from “localized” to “extended” at a value of the strength of the off-diagonal term that depends on energy, in contrast to the Harper equation where the transition is energy-independent. This has a crucial effect on the measure of the spectrum.