BEHAVIOR OF ADIABATIC POTENTIALS IN THE SCATTERING OF 3 PARTICLES

We examine, and discuss, the basic procedure (the "trick" needed in 1, 2 or 3 dimensions) to calculate 3 to 3 scattering using hyperspherical harmonics. To illustrate this, we look at three particles in a plane, subject to finite repulsive core interactions, and obtain, numerically, long-range adiabatic effective potentials. Their asymptotic behaviour is such as to provide the correct (non-analytic) low-energy dependence of the three-body phase shifts. We note that the expansion is not uniform in the number of harmonics needed as the hyperradius changes, and that the asymptotic behaviour of the converged effective potentials differs from that of the matrix elements.