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

TL;DR

This paper compares the supercell and superspace methods for approximating the spectra of quasiperiodic differential operators, demonstrating their convergence and illustrating with Schrödinger and Helmholtz examples.

Contribution

It provides a convergence analysis of two popular methods for spectral computation of quasiperiodic operators, connecting them through Floquet-Bloch theory.

Findings

Both methods converge to the true spectrum.

Illustrations include Schrödinger and Helmholtz operators.

Theoretical insights link supercell and superspace approaches.

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 Schr\"odinger and Helmholtz operators.