Reduced projection method for quasiperiodic Schrödinger eigenvalue problems
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This paper presents a reduced algorithm to the classical projection method for the solution of d-dimensional quasiperiodic problems, particularly Schrödinger eigenvalue problems. Using the properties of the Schrödinger operator in higher-dimensional space via a projection matrix of size d× 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-dD^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 Schrödinger eigenvalue problems in one and two dimensions to demonstrate the accuracy and efficiency of our proposed method.
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