Embree, Mark2013-09-162013-09-162013-09-162013-09-162013-052013-09-16May 2013Puelz, Charles. "Improved Spectral Calculations for Discrete Schroedinger Operators." (2013) Master’s Thesis, Rice University. <a href="https://hdl.handle.net/1911/72024">https://hdl.handle.net/1911/72024</a>.https://hdl.handle.net/1911/72024This 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.application/pdfengCopyright is held by the author, unless otherwise indicated. Permission to reuse, publish, or reproduce the work beyond the bounds of fair use or other exemptions to copyright law must be obtained from the copyright holder.QuasicrystalSchrödinger operatorsDiscrete Schroedinger operatorJacobi operatorCantor spectrumFractal dimentionLarge scale eigenvalue computationsThesis2013-09-16123456789/ETD-2013-05-444