Off-ShellTMatrix and the Jost Function

Abstract

An off-shell effective-rangelike theory is developed for the low-energy two-nucleon $T$ matrix. It is shown that the parameters in the theory can be determined from on-shell scattering data. The parameters are determined from the ${}_{}{}^3{}_{}{}^{}S_1^{}{}_{}{}^{}$ and ${}_{}{}^1{}_{}{}^{}S_0^{}{}_{}{}^{}$ two-nucleon phase shifts, and the validity of the off-shell formula is tested by means of examples. For the cases considered, the formula is found to work very well. A new proof for the separability of the two-body $T$ matrix near bound-state and resonance energies is given. This proof is based on the properties of the $T$ matrix in configuration space. A theorem on the factorization of the Jost function is developed and used to solve the inversion problem for rank-two separable potentials. Results are given for phase shifts that become hard-core phase shifts at high energies and for tensor forces. These results allow one to construct a separable potential that has the same on-shell $T$ matrix and bound-state wave function as a realistic local potential.