Separable Potentials Utilizing Resonant States

Abstract

A method is developed for treating states corresponding to poles at unphysical energies in the two-particle $S$-matrix element for potential scattering. With this method, resonance and bound states can be considered on an equal footing. An inner product of such states is introduced, in terms of which these state are found to form an orthonormal set. An approximation method based on the construction of a separable potential using resonant and bound states is introduced for obtaining matrix elements of two-particle operators. Using the extended inner product, it is shown that the poles in the matrix elements of two-particle operators can be isolated by the approximation method. $S$-matrix elements and diagonal elements of the partial-wave Green's function for a square-well potential are calculated, using the approximation method, and compared with the exact quantities. Very good agreement is found. The approximation has application in the shell-model approach to nuclear reactions and to the three-body problem.