Global scaling properties of the spectrum for a quasiperiodic schrödinger equation

A tight-binding model in one dimension with an incommensurate potential Vₙ=λcos(2πσn) is investigated. It is found that at the critical point of the localization transition λ=2, there is a finite range of scaling indices αₘᵢₙ≤α≤αₘₐₓ each of which is associated with a fractal dimension f(α). In the extended region 0<λ<2, scaling is "trivial" with a single index α=1 almost everywhere in the spectrum, while in the localized region λ>2, there is no scaling.