Cantor spectra and scaling of gap widths in deterministic aperiodic systems

The relationship between geometry and physical properties of aperiodic structures is investigated by considering the example of the tight-binding Schr\"odinger 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.