Solving the bound-state Schrodinger equation by reproducing kernel interpolation

Based on reproducing kernel Hilbert space theory and radial basis approxima tion theory, a grid method is developed for numerically solving the N-dimen sional bound-state Schrodinger equation. Central to the method is the const ruction of an appropriate bounded reproducing kernel (RK) Lambda(alpha)(par allel to r parallel to) from the linear operator -del(r)(2) + lambda(2) whe re del(r)(2) is the N-dimensional Laptacian, lambda>0 is a parameter relate d to the binding energy of the system under study, and the real number alph a>N. The proposed (Sobolev) RK Lambda(alpha)(r,r') is shown to be a positiv e-definite radial basis function, and it matches the asymptotic solutions o f the bound-state Schrodinger equation. Numerical tests for the one-dimensi onal (ID) Morse potential and 2D Henon-Heiles potential reveal that the met hod can accurately and efficiently yield all the energy levels up to the di ssociation limit. Comparisons are also made with the results based on the d istributed Gaussian basis method in the 1D case as well as the distributed approximating functional method in both 1D and 2D cases.