Solving the bound-state Schrödinger equation by reproducing kernel interpolation

Based on reproducing kernel Hilbert space theory and radial basis approximation theory, a grid method is developed for numerically solving the N-dimensional bound-state Schrödinger equation. Central to the method is the construction of an appropriate bounded reproducing kernel (RK) Λα(r) from the linear operator -∇r2+λ2 where ∇r2 is the N-dimensional Laplacian, λ>0 is a parameter related to the binding energy of the system under study, and the real number α>N. The proposed (Sobolev) RK Λα(r,r′) is shown to be a positive-definite radial basis function, and it matches the asymptotic solutions of the bound-state Schrödinger equation. Numerical tests for the one-dimensional (1D) Morse potential and 2D Henon-Heiles potential reveal that the method can accurately and efficiently yield all the energy levels up to the dissociation limit. Comparisons are also made with the results based on the distributed Gaussian basis method in the 1D case as well as the distributed approximating functional method in both 1D and 2D cases. ©2000 The American Physical Society.