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Convergence of supercell and superspace methods for computing spectra of quasiperiodic operators

by B. Davies and C. Thalhammer

(Report number 2025-03)

Abstract
We study the convergence of two of the most widely used and intuitive approaches for computing the spectra of differential operators with quasiperiodic coefficients: the supercell method and the superspace method. In both cases, Floquet-Bloch theory for periodic operators can be used to compute approximations to the spectrum. We illustrate our results with examples of Schrodinger and Helmholtz operators.

Keywords: quasicrystal, cut and project, fractal spectrum, Cantor set, Fibonacci tiling, almost Mathieu operator

@Techreport{DT25_1124,
  author = {B. Davies and C. Thalhammer},
  title = {Convergence of supercell and superspace methods for computing spectra of quasiperiodic operators},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2025-03},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2025/2025-03.pdf },
  year = {2025}
}

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