Spectral splitting and wave-function scaling in quasicrystalline and hierarchical structures

We give a comprehensive presentation of a renormalization-group theory for the study of a set of one-dimensional Schr\"odinger equations on quasicrystalline and hierarchical structures. Particular attention is focused on the spectral clustering and wave-function scaling properties. New results are given on (i) a general characterization of the wave functions, (ii) the scaling of the localized edge states, and (iii) a hierarchical-lattice implementation of the renormalization group.