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 Schr\"{o}dinger equation.

翻译：准周期系统是一类重要的空间填充有序结构，不具有衰减性和平移不变性。如何精确高效地求解准周期系统是极具挑战性的课题。投影法（PM）[J. Comput. Phys., 256: 428, 2014]作为一种有效方案被提出并用于准周期系统计算，众多研究表明该方法具有高精度和高效率。然而，目前缺乏对PM的严格理论分析。本文通过建立准周期函数及其高维周期函数的数学框架，对PM进行了严格的收敛性分析，并基于该框架给出了准周期谱方法（QSM）的理论分析。结果表明PM与QSM均具有指数衰减特性，且QSM（PM）是周期傅里叶谱方法（拟谱方法）的推广。在此基础上比较了PM与QSM计算准周期系统的计算复杂度，其中PM可应用快速傅里叶变换而QSM不能。最后，本文研究了PM、QSM及周期近似法在求解线性含时准周期薛定谔方程时的精度与效率。