A discrete method of constructing central and angular-momentum dependent potentials

Abstract

For a central potential the Schrödinger equation for the lth partial wave is first transformed to a differential equation which depends linearly on l, and then the latter form is approximated by a difference equation. The structure of this difference equation is such that it makes it possible to arrive, in a natural way, at a Thiele's interpolation formula for the logarithmic derivative of the wave function outside the potential, which is related to the discrete form of the [Formula: see text]-matrix. If the numerical values of the [Formula: see text]-matrix, at a given energy, are known for N partial waves, then the found interpolation formula can be used to determine the potential at N equidistant points. This method, in a sense, complements the Newton–Sabatier method since it works better for low energy scattering. It can easily be generalized and applied to the problems where the potential depends on angular momentum, the example of inverse problem for spin-orbit interaction has been studied as an application of this method. Numerical results are given for a number of potential models.