Three-Body Equations and the Moller Operator Approach

Abstract

A family of three-body equations are investigated in relation with the M¢ller operator approach (MOA) of Sandhas. The family consists of a single three-body Lippmann-Schwinger (LS) equation (the LS equation), the LS triad of Glockle, the Faddeev equation, and the Faddeev·equivalent LS triad (the FE-LS triad). We show that all of these equations can be derived from the MOA. Ho:wever, of these, the LS equation and the LS triad that are derived without introducing the Faddee~ decomposition of the total Green function (the FDG) have integral kernels that are either non-compact (the LS equation) or not defined uniquely (the LS triad). We show that, to derive the Faddeev equation and the FE-LS triad from the MOA, it is necessary to incorporate the FDG. This implies that the consideration of the projection properties of the M¢l1er operators (PPMO) among three-body Hilbert spaces alone is not sufficient to assure the equivalence of the MOA to the Faddeev theory. However, we also confirm that, with or without the FDG, all the relations among scattering states obtained from the PPMO are consistent with the three-body equations that are pertinent to the treatment.