An efficient spectral-Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrödinger equations

In this paper, we propose an efficient spectral-Galerkin method based on a dimension reduction scheme for eigenvalue problems of Schrodinger equations. Firstly, we carry out a truncation from a three-dimensional unbounded domain to a bounded spherical domain. By using spherical coordinate transformation and spherical harmonic expansion, we transform the original problem into a series of one-dimensional eigenvalue problem that can be solved effectively. Secondly, we introduce a weighted Sobolev space to treat the singularity in the effective potential. Using the property of orthogonal polynomials in weighted Sobolev space, the error estimate for the approximate eigenvalues and corresponding eigenfunctions are proved. Error estimates show that our numerical method can achieve spectral accuracy for approximate eigenvalues and eigenfunctions. Finally, we give some numerical examples to demonstrate the efficiency of our algorithms and the correctness of the theoretical results.