Alternative interpretations of the many-particle Lippmann-Schwinger equation.

Abstract

Possible alternative interpretations of the Lippmann-Schwinger integral equation for multiparticle (ng2) systems are investigated and are shown to be equivalent if integrals which occur are uniformly convergent, as is reasonable. At real energies E, the derivation of the Lippmann-Schwinger equation from the Schr\odinger equation involves various surface integrals at infinity in configuration space. It is shown that the values of these surface integrals are related to the values of certain volume integrals at complex energies (E+i\ensuremath{\epsilon}) in the limit \ensuremath{\epsilon}\ensuremath{\rightarrow}0, originally examined by Lippmann. It is further proved that a number of these surface integrals vanish together, a result which---though plausible---previously had to be assumed. The results of this paper confirm previous studies showing that the solutions to the multiparticle Lippmann-Schwinger equation need not be unique. Because of certain convergence difficulties which can occur, the analysis of this paper is not wholly valid for ``three-body'' collisions (defined as collisions involving three independently incident aggregates of the fundamental particles comprising the multiparticle system), or for the even more complicated collisions involving ng3 incident aggregates.