In this paper, we propose a new algorithm, the irrational-window-filter projection method (IWFPM), for quasiperiodic systems with concentrated spectral point distribution. Based on the projection method (PM), IWFPM filters out dominant spectral points by defining an irrational window and uses a corresponding index-shift transform to make the FFT available. The error analysis on the function approximation level is also given. We apply IWFPM to 1D, 2D, and 3D quasiperiodic Schr\"odinger eigenproblems (QSEs) to demonstrate its accuracy and efficiency. IWFPM exhibits a significant computational advantage over PM for both extended and localized quantum states. More importantly, by using IWFPM, the existence of Anderson localization in 2D and 3D QSEs is numerically verified.

翻译：本文针对谱点分布集中的准周期系统，提出了一种新算法——无理窗滤波投影法（IWFPM）。该方法基于投影法（PM），通过定义一个无理窗来滤除主导谱点，并采用相应的指标平移变换以实现快速傅里叶变换（FFT）。文中同时给出了函数逼近层面的误差分析。我们将IWFPM应用于一维、二维和三维准周期薛定谔本征问题（QSEs），验证了其精度与效率。对于扩展态与局域化量子态，IWFPM均展现出显著优于PM的计算优势。更重要的是，通过应用IWFPM，我们在数值上验证了二维与三维QSEs中安德森局域化现象的存在性。