Quasiperiodic Elliptic Operators: Projection Method and Convergence Analysis

TL;DR

This paper introduces a high-accuracy projection method for computing eigenpairs of quasiperiodic elliptic operators, overcoming challenges related to spectral structure and convergence in non-compact settings.

Contribution

It develops a novel numerical approach embedding QEOs into higher-dimensional tori and provides rigorous convergence analysis with spectral accuracy guarantees.

Findings

Method achieves spectral accuracy in numerical experiments.

Successfully computes eigenpairs for 1D, 2D, and 3D QEOs.

Validates effectiveness with photonic quasicrystal models.

Abstract

Quasiperiodic elliptic operators (QEOs) serve as fundamental models in both mathematics and physics, as exemplified by their role in the numerical modeling of one-dimensional photonic quasicrystals. However, distinct from periodic elliptic operators, approximating eigenpairs for QEOs poses significant challenges, particularly in capturing the full spectral structure (notably the continuous spectrum) and deriving convergence guarantees in the absence of compactness. In this paper, we develop a high-accuracy numerical method to compute eigenpairs of QEOs based on the projection method, which embeds quasiperiodic operators into a higher-dimensional periodic torus. To address the non-compactness issue, we construct a directional-derivative Hilbert space along irrational manifolds of a high-dimensional torus and characterize operators equivalent to QEOs within this space. By integrating spectral theory for non-compact operators into the Babu\v{s}ka-Osborn eigenproblem framework, we establish rigorous convergence analysis and prove that our method achieves spectral accuracy. Numerical experiments validate the accuracy and efficiency of the proposed method, including a one-dimensional photonic quasicrystal and two- and three-dimensional QEOs.