Reduced Projection Method for Photonic Moiré Lattices

This paper presents a reduced projection method for the solution of quasiperiodic Schrödinger eigenvalue problems for photonic moiré lattices. Using the properties of the Schrödinger 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 moiré lattices in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.