Elastically Equivalent Potentials

We define a nondenumerably infinite class of two-body potentials that give $T$ matrix elements that are identical on the energy shell but are different off shell. The existence of this class is shown to be related to a nonuniqueness in the off-shell continuation of the Schrödinger equation. A relation for the difference between off-shell elements of $T$ matrices that are equal on shell is given in a form that is convenient for the systematic study of off-shell effects in systems of more than two nucleons. In particular, these relations allow one to investigate subclasses of elastically equivalent interaction potentials that satisfy restrictions such as finite range. The possibility that elastically equivalent interactions have different off-shell symmetry properties is discussed, and some observations concerning the use and interpretation of elastically equivalent potentials in many-body calculations are presented.