Spectrum of a one-dimensional hierarchical model

The spectrum of a discrete Schroedinger operator with a hierarchically distributed potential is studied both by a renormalization group technique and by numerical analysis. A suitable choice of the potential makes it possible to reduce the original problem to a two-dimensional map. Scaling laws for the band-edge energy and for the integrated density of states are predicted together with the global properties of the spectrum. Different scaling regimes are obtained depending on a hierarchy positive parameter R: for R < 1/2 the usual scaling laws for the periodic case are obtained, while for R>1/2 the scaling behavior depends explicitly on R