Connection Between theS-Matrix and the Tensor Force

The present paper is a study of the radial Schrödinger equations for the case of an interaction between the $l\mathrm{th}$ and the $\mathrm{}(l+2\mathrm{})\mathrm{}\mathrm{nd}$ angular momentum, produced by the tensor force in the presence of spin-orbit coupling. It contains a number of theorems, known for central potentials, concerning the low-energy behavior of the $S$-matrix, bound states, and zero-energy resonance. The construction, in two stages, of all potentials belonging to a given $S$-matrix and given bound states, is described. The step from the spectral function to the potential involves the generalization of the Gel'fand Levitan equation given in a recent paper; that from the $S$-matrix to the spectral function, a procedure due to Plemelj also outlined in that paper. The latter procedure leads to a restriction on the $S$-matrix necessary for a short range potential to exist. If there is such a potential, it is uniquely determined by the $S$-matrix, the binding energies, and as many real, symmetric, positive semidefinite matrices as there are bound states.