On the approximate solution through continued fractions of the schrödinger equation with central potentials for positive energies

We present an extension to positive energies of an approximation method based on continued fractions already used to provide approximate energies and wave functions for bound states. A formalism particularly adapted for numerical computation is proposed and its applications are shown for the free particle case and short and long range central potentials, for the lowest angular momenta. The domain of convergence is displayed; especially for low and medium energies, the results are quite good. Some care is devoted to the successful determination of theS-wave scattering length for the Yukawa potential. The mathematical relationship with Calogero's formulation of the problem is established.