Asymptotic Completeness for Four-Body Schrödinger Operators with Short-Range Interactions

Abstract

The present work is a continuation to the previous one [15] in which the author has studied the problem of asymptotic completeness in the case of threebody systems with short-range pair interactions with a view to making transparent the proof of asymptotic completeness in the remarkable work by SigalSoffer [13] which deals with the case of general TV-body systems. The proof in [15] has been in principle based on the same idea as in [13] but several new ingredients have been added to the techniques developed there. In particular, the proof does not have required a phase space partition of unity with the property that the boundaries of its support lie in the classically forbidden region. The construction of such a phase space partition of unity is one of the most essential steps in the original proof by [13], The aim of this work is to develope further the argument used in [15] to prove the asymptotic completeness for four-body systems with short-range pair interactions. The author hopes that the previous and present works reveal the difficulties to be overcome in the future study towards proving the asymptotic completeness for general Nbody, 7V;>5, systems.The precise formulation of the obtained result requires several complicated but basic notations and definitions in many-body scattering theory.For notational brevity, we here consider only a simple system of four particles with identical masses normalized by nij=l, l^j'^4, moving in the three-dimensional space R z . For such a system, the configuration space X in the center of mass frame is given by C=^r=tri, r 2 , r a , ., *• 3 -with the usual scalar product (0.1) r-f=SJ-irrĈ