The spectrum of a one-dimensional hierarchical model

The spectrum of a discrete Schrödinger 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 energyE be and for the integrated density of states eegr are predicted together with the global properties of the spectrum. Different scaling regimes are obtained depending on a hierarchy positive parameterR: for R<1 the="" usual="" scaling="" laws="" for="" the="" periodic="" case="" are="" obtained,="" while="">R>1/2 the scaling behavior depends explicitly onR.