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：准周期系统是具有空间填充有序结构的重要系统，既无衰减性也不具备平移不变性。如何精确高效地求解准周期系统是一个重大挑战。投影法（PM）[J. Comput. Phys., 256: 428, 2014]作为一种有效方法被提出用于计算准周期系统。多项研究表明，PM是求解准周期系统精确且高效的方法，然而目前缺乏对PM的理论分析。本文通过建立准周期函数及其高维周期函数的数学框架，给出了PM的严格收敛性分析。基于该框架，我们同时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（伪谱）方法的推广。在此基础上，我们分析了PM与QSM在计算准周期系统时的计算复杂度——PM可采用快速傅里叶变换，而QSM则无法实现。此外，我们进一步研究了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。