Low-Energy Three-Body Scattering in a Hyperspherical Adiabatic Basis

We present a general formalism for obtaining low energy expansions of three-body scattering matrices for finite range, repulsive potentials. The formalism involves several steps: first the Schrodinger equation is written in hyperspherical coordinates; the wave function is then expanded in a hyperspherical harmonic basis; finally we pass to an adiabatic basis and solve (for low energies) the resulting set of coupled equations using an iterative Green's function approach. This is made possible by the fact that the coupled equations in this basis decouple for large values of the hyper-radius as E to 0. An accurate wave function, valid at low energies, can therefore be found to start the iteration. The resulting iteration integrals are then obtained analytically. We apply this formalism to a three-body system for which exact solutions are known. It is a model of McGuire's which involves three-particles in one-dimension interacting via delta-function potentials. We note that we obtain exact results for the first three terms in the expansion of the scattering matrix for the model.