Numerical Methods and Analysis of Computing Quasiperiodic Systems

TL;DR

The paper tackles the challenge of numerically solving quasiperiodic systems, which lack translational invariance, by developing a rigorous framework that connects a quasiperiodic function $f(\mathbf{x})=F(\mathbf{P}^T\mathbf{x})$ to a high-dimensional periodic parent function $F$ on a torus. It analyzes two numerical schemes, the quasiperiodic spectral method (QSM) and the projection method (PM), showing that both achieve exponential convergence by exploiting this parent–child relationship; PM additionally leverages FFT via discrete Fourier–Bohr transforms. The authors prove that the quasiperiodic Fourier coefficients equal their parent Fourier coefficients ($\hat{f}_{\mathbf{k}}=\hat{F}_{\mathbf{k}}$) using ergodic theory, and provide $L^2$ and $L^\infty$ error bounds for QSM and PM, alongside complexity comparisons: PM can use FFT while QSM cannot. They apply these methods to the linear time-dependent quasiperiodic Schrödinger equation (TQSE), demonstrating that PM and QSM offer superior accuracy and efficiency compared to periodic approximation methods, with PM achieving practical robustness and scalability. Overall, the work delivers the first theoretical guarantees for PM and positions PM and QSM as powerful, complementary tools for simulating a broad class of quasiperiodic systems.

Abstract

Quasiperiodic systems are important space-filling ordered structures, without decay and translational invariance. How to solve quasiperiodic systems accurately and efficiently is of great challenge. A useful approach, the projection method (PM) [J. Comput. Phys., 256: 428, 2014], has been proposed to compute quasiperiodic systems. Various studies have demonstrated that the PM is an accurate and efficient method to solve quasiperiodic systems. However, there is a lack of theoretical analysis of PM. In this paper, we present a rigorous convergence analysis of the PM by establishing a mathematical framework of quasiperiodic functions and their high-dimensional periodic functions. We also give a theoretical analysis of quasiperiodic spectral method (QSM) based on this framework. Results demonstrate that PM and QSM both have exponential decay, and the QSM (PM) is a generalization of the periodic Fourier spectral (pseudo-spectral) method. Then we analyze the computational complexity of PM and QSM in calculating quasiperiodic systems. The PM can use fast Fourier transform, while the QSM cannot. Moreover, we investigate the accuracy and efficiency of PM, QSM and periodic approximation method in solving the linear time-dependent quasiperiodic Schrödinger equation.

Figures (4)

• Figure 1: The process of modulo a two-dimensional irrational slice $\bm P^T x$ where $\bm P=(1,\sqrt{3})$, $x\in\mathbb{R}$.
• Figure 1: Approximation error of PAM as the domain size $L$ increases.
• Figure 2: The relationship between the numerical error $e_N$ and $N$.
• Figure 3: The relationship between the numerical error $e_N$ and CPU time (s) when $N=2,4,8,16,32,$ respectively.

Theorems & Definitions (29)

• Definition 2.1
• Theorem 2.2
• Remark 3.1
• Theorem 4.1
• Definition 4.2
• Definition 4.3
• Lemma 1
• Theorem 4.4
• Proposition 2
• Proof 1