Generalized Faddeev Equations for N-Particle Scattering

A proposal is made for reducing the solution of the N-particle Lippmann-Schwinger equation to that of smaller sets of particles. This consists of first writing the N-particle equation in terms of all possible $N/2$-particle Lippmann-Schwinger equations. (If N is odd this needs a minor modification.) The second step requires a decoupling of the resolvents for the fewer particle systems so that each can be solved separately. This generalization of the Faddeev approach deals only with connected kernels and the homogeneous solution reproduces the N-particle Schr\"odinger equation. For four particles the proposed method involves only a $3\times3$ matrix whereas other approaches typically require the solution of at least a $7\times7$ matrix equation.