| Summary: | This work details an O(n^2) algorithm for computing the spectra of discrete Schroedinger
operators with periodic potentials. Spectra of these objects enhance our understanding of fundamental aperiodic physical systems and contain rich theoretical structure
of interest to the mathematical community. Previous work on the Harper model led
to an O(n^2) algorithm relying on properties not satisfied by other aperiodic operators. Physicists working with the Fibonacci Hamiltonian, a popular quasicrystal
model, have instead used a problematic dynamical map approach or a sluggish O(n^3)
procedure for their calculations. The algorithm presented in this work, a blend of well-established eigenvalue/vector algorithms, provides researchers with a more robust computational tool of general utility. Application to the Fibonacci Hamiltonian
in the sparsely studied intermediate coupling regime reveals structure in canonical
coverings of the spectrum that will prove useful in motivating conjectures regarding
band combinatorics and fractal dimensions.
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