Note on Schwinger's Variational Method

Abstract

It is shown that one of the variational methods of Schwinger applied to the one-dimensional wave equation gives a lower bound of the absolute value of the scattering phase provided the potential is semi-definite and too strong, a condition satisfied for potentials proposed for the neutron-proton scattering. Further a simple modification of Schwinger's iteration procedure is proposed, which converges not only mathematically but also more rapidly than the original one. An upper bound of the absolute value of the phase is also shown to be obtained if the iteration method is carried out to the second approximation. Numerical examples are given in the case of a square-well potential, and very accurate upper and lower bounds of the phases are obtained.