The transition and the Green operators of an interacting N body system are obtained from the solutions of the N-M body problem where M=1,2,cdots, N-2. This is achieved via the development of a cumulative, non-perturbative approach that makes use of existing knowledge on the system when the number of interacting particles is reduced. The method is applied to four interacting Coulomb particles where the Green operator is expressed as a sum of Green operators of all three body subsystems that can be combined within the four body system. The calculated four particle continuum spectrum is in a remarkable agreement with recent experimental findings. ©2000 The American Physical Society