Cantor spectra and scaling of gap widths in deterministic aperiodic systems

Abstract

The relationship between geometry and physical properties of aperiodic structures is investigated by considering the example of the tight-binding Schrödinger equation in one dimension, where the site potentials are given by an arbitrary deterministic aperiodic sequence. In a perturbative analysis of the integrated density of states, the gaps in the energy spectrum can be ‘‘labeled’’ by the singularities of the Fourier transform of the sequence of potentials. This approach confirms known properties of quasiperiodic and almost-periodic systems, and suggests an extension of them to more general sequences, such as those with a singular continuous Fourier transform. There is strong evidence that the spectrum is a Cantor set with zero measure for a much larger class of models than quasiperiodic ones. The dependence of the widths of various gaps on the potential strength is also determined: several different kinds of behavior are obtained, such as a power law with a nontrivial exponent, or an essential singularity. These general results are compared with those of various other approaches for four self-similar sequences generated by substitution, namely the Thue-Morse sequence, the period-doubling sequence, a ‘‘circle sequence,’’ and the Rudin-Shapiro sequence.