Abstract
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 η are predicted together with the global properties of the spectrum. Different scaling regimes are obtained depending on a hierarchy positive parameterR: for R<1/2 the usual scaling laws for the periodic case are obtained, while forR>1/2 the scaling behavior depends explicitly onR.
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Livi, R., Maritan, A. & Ruffo, S. The spectrum of a one-dimensional hierarchical model. J Stat Phys 52, 595–608 (1988). https://doi.org/10.1007/BF01019719
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DOI: https://doi.org/10.1007/BF01019719