Hierarchical clustering in the spectra of incommensurate systems

Using degenerate perturbation theory to all orders the authors derive the Hofstadter cluster rules (1976) for the spectrum of Harper's equation (1955). They also extend their method to calculate the fractal dimension, D f , of the spectrum and speculate that D f =1/2 for typical values of the incommensurability parameter phi . The clustering rules are also shown to be valid for a one-dimensional model of quasicrystals.