Generalization of Levinson's Theorem to Three-Body Systems

Levinson's theorem is generalized to systems of three particles. The usual two-body result relates the number of bound states of given angular momentum to the corresponding eigenphase shifts of the $S$ matrix. Because of disconnected diagrams the three-body $S$ matrix has continuous eigenphase shifts in addition to any discrete ones; however, it is possible to define a unitary connected matrix that has only discrete eigenphase shifts. Levinson's theorem is given in terms of these phase shifts, and it is the same as the usual multichannel result, except that there are an infinite number of eigenphase shifts to be summed over for each value of the total angular momentum. The proof is carried out within the framework of the Faddeev equations by generalizing Jauch's proof for two-body systems.