This paper presents a reduced projection method for the solution of quasiperiodic Schr\"{o}dinger eigenvalue problems for photonic moir\'e lattices. Using the properties of the Schr\"{o}dinger operator in higher-dimensional space via a projection matrix, we rigorously prove that the generalized Fourier coefficients of the eigenfunctions exhibit faster decay rate 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 significantly. Rigorous error estimates of the proposed reduced projection method are provided, indicating that a small portion of the degrees of freedom is sufficient to achieve the same level of accuracy as the classical projection method. We present numerical examples of photonic moir\'e lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.

翻译：本文提出了一种用于求解光子莫尔晶格准周期薛定谔特征值问题的约化投影方法。利用更高维空间中薛定谔算子通过投影矩阵的性质，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具有更快的衰减速率。随后提出了一种高效的基空间约化策略，显著降低了自由度。我们提供了所提出的约化投影方法的严格误差估计，表明仅需少量自由度即可达到与传统投影方法相同的精度。通过一维和二维光子莫尔晶格的数值算例，验证了该方法的准确性与效率。