Band spectrum of the octagonal quasicrystal: Finite measure, gaps, and chaos.

Abstract

In this paper, we present numerical calculations of the band spectrum of periodic approximants of the octagonal quasiperiodic tiling. We have studied a one-parameter family of Hamiltonians, including the pure hopping case, the Laplacian, and a regime in which atomic potentials prevail. We have found a parameter range over which the spectrum consists of only one band. Moreover, the Lebesgue measure is finite except maybe in a small domain of the parameter range. We have also studied the statistics of the levels. We have obtained a transition between a regime of level repulsion (quantum chaos) and another of level clustering (Cantor spectrum).