Non-Compactness of the Kernel of a 3-Body Lippmann-Schwinger Equation in the Absence of Rearrangement Channels

Abstract

The question of whether or not a single 3-body Lippmann-Schwinger (LS) equation can yield a unique solution in the absence of rearrangement channels (RC) is re-investigated. For a Hermitian Hamiltonian, the non-compactness of the integral kernel is shown by demonstrating the existence of continuous spectrum in a manner of Weinberg. This implies that its cause is the disconnected diagrams present with or without RC. Therefore, even in the absence of RC, either the Faddeev equation or the Faddeev-equivalent LS triad is necessary for a rigorous treatment of a 3-body scattering problem. The case with energy-independent absorbing potentials (EIAP) for 2-body interactions in RC is also discussed. The above non-compactness proof is seen to be applicable for this case as well, provided that at least oI).e of the RC potentials is chosen to be a real potential multiplied by a complex constant. Thus, the apparent uniqueness of the CDCC solutions cannot ab initio be attributed to the use of EIAP in RC.