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Quasiperiodic Elliptic Operators: Projection Method and Convergence Analysis

Kai Jiang, Jiaqi Tang, Qilong Zhai, Qi Zhou

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 Babuška-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.

Quasiperiodic Elliptic Operators: Projection Method and Convergence Analysis

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 Babuška-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.

Paper Structure

This paper contains 16 sections, 15 theorems, 121 equations, 5 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Background
  3. Numerical progress and challenges
  4. Contribution
  5. Organization
  6. Notations
  7. Preliminaries
  8. Quasiperiodic functions
  9. Besicovitch quasiperiodic function spaces
  10. Numerical method and theoretical framework for QEOs
  11. The eigenproblem of QEOs
  12. Discretize QEOs by PM
  13. Convergence analysis
  14. Error estimates
  15. Numerical experiments
  16. ...and 1 more sections

Key Result

Lemma 2.1

For a given continuous quasiperiodic function $f(\bm{x}) = F(\bm{P}^{T}\bm{x}),~\bm{x} \in \mathbb{R}^{d},$ where $F(\bm{y})\in\mathcal{C}(\mathbb{T}^n)$ is the parent function and $\bm{P}$ is the projection matrix, we have $\hat{f}_{\bm{\lambda_{k}}} = \hat{F}_{\bm{k}},~\forall\,\bm{k}\in\mathbb{Z}

Figures (5)

  • Figure 1: Errors of the first two eigenpairs when using PM to solve Example 4.1.
  • Figure 2: Localized and extended states of Example 4.1 solved by PM.
  • Figure 3: Diophantine approximation error $e_{d}$ when using PAM to solve Example 4.1.
  • Figure 4: Comparison of PM and PAM when solving Example 4.1.
  • Figure 5: Errors of the first two eigenpairs when using PM to solve Example 4.2 with $\beta = 0.5\pi,~\theta = 0.2 \pi$.

Theorems & Definitions (31)

  • Definition 2.1
  • Definition 2.2
  • Lemma 2.1: jiang2024, Theorem 2.3
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 21 more