Unitarity and off-shell effects in the impulse approximation

For elastic scattering of a particle from a finite many-body target, the procedure of eliminating the two-body potential in favor of a two-body scattering matrix is examined. It is shown that the theory of Kerman, McManus, and Thaler leads to unitary relations which are not of a particularly convenient form. A simple procedure which identifies the appropriate off-shell scattering matrix to be used in the impulse approximation is presented. The projection operator technique is employed to define an optical potential and then, in a truncated Hilbert space, a relation similar to a two-body Low equation is used to identify the appropriate two-body $T$ matrix. With this latter procedure the unitarity relations are maintained in a form which allows them to be used as a guide to the validity of the impulse approximation.