Highly accurate calculation of higher energy eigenvalues for the radial Schrödinger eigenproblems

In this paper, we derive an efficient numerical scheme for approximating the energy eigenvalues and corresponding wavefunctions of the radial Schrödinger eigenproblems defined on a semi-infinite domain for arbitrary values of the angular momentum number. The numerical scheme is based on the Chebyshev spectral collocation differentiation matrix method. In this scheme, at first, we redefine the radial Schrödinger eigenproblem on a finite interval by adopting an appropriate change of the independent variable. Then, by expanding the wavefunctions of the problem as a series of the Chebyshev polynomials of the first kind, as well as employing the differentiation matrices in order to determine the derivatives of Chebyshev polynomials at the collocation points, considered problem becomes a generalized eigenvalue problem. The convergence behavior and excellent performance of the proposed technique are illustrated through some numerical experiments of the most significant potentials in quantum mechanics. Compared with exact solutions (when available), and those reported previously in the literature, the current scheme achieves higher accuracy and efficiency even when high-index energy eigenvalues are computed.