Investigation of the Hylleraas Method for Determining the Potential-Energy Function from the Phase Shift

The Hylleraas solution of the quantum mechanical inverse problem of scattering theory is discussed. A practical method for deducing the potential from the phase shift by this method and a variant upon it is described. A simple example (a model potential function) is used to illustrate the procedure. For this case, first-order results are obtained analytically, higher-order results by (extensive) numerical computations. The method converges properly if there are no bound states. However, for the model problem under investigation, the procedure converges to yield a phase-equivalent potential when the potential well contains a bound state.