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.

Es werden spektrale Eigenschaften des zweidimensionalen Schrödinger-Operators mit einem zweifach periodischen Potential und starkem magnetischem Feld untersucht mit Hilfe semiklassischer Methoden. Man beschreibt die spektrale Asymptotik durch Benutzung der Reeb-Graph-Technik. Im Falle des rationalen Flusses konstruiert man semiklassische Magneto-Bloch-Funktionen und beschreibt die Asymptotik des Spektrums auf dem physikalischen Beweisniveau. / Spectral properties of the two-dimensional Schroedinger operator with a two-periodic potential and a strong uniform magnetic field is studied with the help of semiclassical methods. The spectral asymptotics is described using the Reeb graph technique. In the case of the rational flux one constructs semiclassical magneto-Bloch functions and describes the asymptotics of the band spectrum on the physical level of proof.