Lippmann's identities in a soluble three-particle model.

Abstract

In some formulations, application of the Lippmann-Schwinger equation to collisions in many-particle (n>2) systems naturally leads to the limits \ensuremath{\epsilon}\ensuremath{\rightarrow}0 of certain volume integrals involving Green's functions at complex energies E+i\ensuremath{\epsilon}. The values of these limits, derived originally by Lippmann using operator techniques, are known as Lippmann's identities. This paper evaluates the limits studied by Lippmann in a one-dimensional three-body model (due to McGuire) which has been much employed recently to test Lippmann-Schwinger formulations of scattering theory; the results verify the Lippmann identities. Also evaluated in the McGuire model are a number of related identities, not due to Lippmann, that have been derived recently via quite conventional mathematical operations which yielded a previously unsuspected relationship between Lippmann's \ensuremath{\epsilon}\ensuremath{\rightarrow}0 limits and a class of surface integrals at infinity in configuration space, originally studied by Gerjuoy. All the results completely confirm the validity of the aforementioned mathematical operations and, by implication, also confirm the disputed conclusion that the many-particle Lippmann-Schwinger equation does not have unique solutions.