Bethe's hypothesis and Feynman diagrams: Exact calculation of a three-body scattering amplitude by perturbation theory

The perturbation series for three-body scattering is discussed and summed to all orders in a nonrelativistic one-dimensional theory with $\ensuremath{\delta}$-function interaction. The summation is accomplished by analyzing three simultaneous integral equations satisfied by the three-body scattering amplitude and appropriately decomposing and recombining the kernels which appear in these equations. The homogeneous integral terms can be entirely eliminated from one equation in favor of the scattering amplitude itself, yielding an algebraic equation which is easily solved. The three-particle incoming wave function in configuration space is constructed from the scattering amplitude and shown to be that given by Bethe's hypothesis.