The structure of a non-relativistic S -matrix

The S-matrix for the non-relativistic scattering of a particle by a target possessing discrete excited states is considered. Assuming the S-matrix to be an analytic function of the energy, the unitarity and reversibility conditions are extended into the complex plane, and it is shown how all the elements of the S-matrix can be constructed from a single function P(k, K, L$\ldots$) of the momenta of the particle in the various channels, which has zeros at the poles of S. For the two channel problem the S-matrix is constructed explicitly from the positions of its poles in the complex plane together with conditions at $\infty$, and necessary conditions on the number and positions of the poles are formulated.