Analytical Expressions for a Hyperspherical Adiabatic Basis Three Particles in 2 Dimensions

For a particular case of three-body scattering in 2 dimensions, we demonstrate analytically that the behaviour of the adiabatic potential is different from that of the hyperspherical coupling matrix elements, thereby leading to a phase shift that tends to zero, as the energy goes to zero, instead of to a constant. We consider two particles interacting with binary repulsive step potentials, one acting as a spectator, and solve analytically for the adiabatic eigenvalues and eigenvectors, for all values of ρ. We are thereby able to obtain the leading terms in the long range behaviour of the effective potentials, and confirm its inverse logarithmic nature, suggested from a previous numerical study.