Computing Eigenvalues and Eigenfunctions Of Schrodinger Equations Using a Model Reduction Approach

Abstract

We present a model reduction approach to construct problem dependent basis functions and compute eigenvalues and eigenfunctions of stationary Schrödinger equations. The basis functions are defined on coarse meshes and obtained through solving an optimization problem. We shall show that the basis functions span a lowdimensional generalized finite element space that accurately preserves the lowermost eigenvalues and eigenfunctions of the stationary Schrödinger equations. Therefore, our method avoids the application of eigenvalue solver on fine-scale discretization and offers considerable savings in solving eigenvalues and eigenfunctions of Schr ödinger equations. The construction of the basis functions are independent of each other; thus our method is perfectly parallel. We also provide error estimates for the eigenvalues obtained by our new method. Numerical results are presented to demonstrate the accuracy and efficiency of the proposed method, especially Schrödinger equations with double well potentials are tested.