Global scaling properties of the spectrum for a quasiperiodic schrödinger equation

Abstract

A tight-binding model in one dimension with an incommensurate potential $V_n=\lambda cos\left(2\mathrm{}\pi \sigma n\right)$ is investigated. It is found that at the critical point of the localization transition $\lambda =2\mathrm{}$, there is a finite range of scaling indices ${\alpha }_{min}\le \alpha \le {\alpha }_{max}$ each of which is associated with a fractal dimension $\mathcal{f}\left(\alpha \right)$. In the extended region $0<\mathrm{}\lambda <2\mathrm{}$, scaling is "trivial" with a single index $\alpha =1\mathrm{}$ almost everywhere in the spectrum, while in the localized region $\lambda >2\mathrm{}$, there is no scaling.