An efficient algorithm for the Schrödinger-Poisson eigenvalue problem

Authors:
S. -L. Chang;C. -S. Chien;B. -W. Jeng
Affiliations:
Center for General Education, Southern Taiwan University of Technology, Tainan 710, Taiwan;Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan;Department of Applied Mathematics, National Chung-Hsing University, Taichung 402, Taiwan
Venue:
Journal of Computational and Applied Mathematics
Year:
2007

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Abstract

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrodinger-Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrodinger eigenvalue problem, and use the block Lanczos method to compute the first k eigenpairs of the Schrodinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrodinger eigenvalue problem are affected by the dopant in the Schrodinger-Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h^3).