This paper presents a reduced algorithm to the classical projection method for the solution of $d$-dimensional quasiperiodic problems, particularly Schr\"{o}dinger eigenvalue problems. Using the properties of the Schr\"{o}dinger operator in higher-dimensional space via a projection matrix of size $d\times n$, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions decay exponentially along a fixed direction associated with the projection matrix. An efficient reduction strategy of the basis space is then proposed to reduce the degrees of freedom from $O(N^{n})$ to $O(N^{n-d}D^d)$, where $N$ is the number of Fourier grids in one dimension and the truncation coefficient $D$ is much less than $N$. Correspondingly, the computational complexity of the proposed algorithm for solving the first $k$ eigenpairs using the Krylov subspace method decreases from $O(kN^{2n})$ to $O(kN^{2(n-d)}D^{2d})$. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small $D$ is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of quasiperiodic Schr\"{o}dinger eigenvalue problems in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.

翻译：本文提出了一种经典投影方法的约化算法，用于求解$d$维准周期问题，特别是薛定谔本征值问题。利用薛定谔算子在更高维空间中的性质，通过大小为$d\times n$的投影矩阵，我们严格证明了本征函数的广义傅里叶系数沿投影矩阵确定的固定方向呈指数衰减。随后提出一种高效的基空间约化策略，将自由度从$O(N^{n})$降低至$O(N^{n-d}D^d)$，其中$N$为一维傅里叶网格数，截断系数$D$远小于$N$。相应地，采用Krylov子空间方法求解前$k$个本征对的算法计算复杂度从$O(kN^{2n})$降至$O(kN^{2(n-d)}D^{2d})$。本文给出了所提出的约化投影方法的严格误差估计，表明较小的$D$即可达到与经典投影方法相同的精度。我们通过一维和二维准周期薛定谔本征值问题的数值算例，验证了所提方法的准确性和高效性。