Efficient Spectral Method For The Eigenvalue Problem Of Schrodinger Equations On Unbounded Domain

An efficient Laguerre-Galerkin spectral method based on a dimen-sion reduction scheme is proposed for eigenvalue problems of Schrodinger equations on unbounded domain.For eigenvalue problems of two-dimensional Schrodinger equation in unbounded domain.First of all,we use the polar coordinate trans-formation and Fourier expansion to reduce the original problem to a sequence of equivalent one-dimensional eigenvalue problems that can be solved individually in parallel,and we derive the pole conditions.Then,for each one-dimensional eigenvalue problem,we introduce a suitable Sobolev space and derive weak form and associated discrete scheme.Then according to the properties of Laguerre polynomials and functions,for m=0 and m?0,we respectively construct a set of appropriate basis functions to translate discrete scheme into linear system of eigenvalue problem.In addition,we prove that the stiffness matrix and mass matrix are all sparse for some typical effective poten-tials.Finally,we give a large number of numerical examples,and the numerical results indicate that our algorithms are very efficient.For eigenvalue problems of three-dimensional Schrodinger equa-tion in unbounded domain.Firstly,by using spherical coordinate transformation and spherical harmonic expansion,the original problem is reduced to a sequence of equivalent one-dimensional eigenvalue prob-lems,so that the singularity of the effective potential can be solved.Secondly,a weighted Sobolev space is introduced to establish weak for-m and associated discrete scheme of this problem.Thirdly,according to the properties of Laguerre polynomials and functions,we construct a set of appropriate basis functions to translate discrete scheme into linear system of eigenvalue problem.In addition,we prove that the stiffness matrix and mass matrix are all sparse for some typical effec-tive potentials.Finally,some numerical experiments are given,and the numerical results show that the algorithm is very stable and accurate.