Lippmann-Schwinger equation in a soluble three-body model: Surface integrals at infinity.

Abstract

At real energies E, the derivation of the Lippmann-Schwinger integral equation from the Schr\"odinger equation involves various surface integrals at infinity in configuration space. Plausible assumptions about the values of these surface integrals made originally by Gerjuoy imply that the many-particle (n>2) Lippmann-Schwinger equation generally has nonunique solutions. This paper evaluates these surface integrals in the same one-dimensional three-body model (of McGuire) employed recently to demonstrate the nonuniqueness explicitly. The computed values of the surface integrals agree precisely with Gerjuoy's hypotheses. These results further confirm the conclusion that the many-particle Lippmann-Schwinger equation has nonunique solutions in actual three-dimensional collisions, and support the belief that the aforesaid derivation of the real energy Lippmann-Schwinger equation is mathematically sound.