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：本文提出一种用于求解光子莫尔晶格准周期薛定谔特征值问题的降维投影方法。通过投影矩阵利用高维空间中薛定谔算子的性质，我们严格证明了特征函数的广义傅里叶系数沿投影矩阵相关固定方向具有更快的衰减速率。随后提出一种高效的基空间约简策略，可显著降低自由度数量。本文给出了所提降维投影方法的严格误差估计，表明仅需少量自由度即可达到经典投影方法同等精度水平。我们通过一维和二维光子莫尔晶格的数值算例，验证了所提方法的精度与效率。