LOW-ENERGY 3-BODY SCATTERING IN A HYPERSPHERICAL ADIABATIC BASIS

We propose a hyperspherical adiabatic formalism for the calculation of the 3-to-3 S-matrix at low energy, for repulsive potentials, and use it then in a model calculation. That is for McGuire's model (3 particl es in one dimension subject to repulsive delta-function interactions), we use analytical expressions for the hyperspherical adiabatic basis, the adiabatic coupling matrix elements, and eigenpotentials to obtain the first terms of the exact S-matrix analytically, in an expansion i n powers of the wave number. We were able to associate the definite po wers of q in the expansion of the S-matrix to the corresponding invers e powers of rho in the expansions of the adiabatic eigenpotentials and coupling matrix elements. We investigate the effect of making the ''u sual'' approximations found in the literature (extreme and uncoupled a diabatic approximations), when calculating the diagonal and off-diagon al S-matrix elements. Finally, we show that the coupled adiabatic equa tions uncouple as the energy goes to zero.