Exact solution for three particles interacting via zero-range potentials

Abstract

Exact solutions for three identical bosons interacting via zero-range s -wave potentials are derived. The solutions are contour integrals over a product of hyperradial Bessel functions times angular functions weighted by a coefficient. The product function is a solution of the free-particle Schrödigner equation and the weight function is chosen to satisfy the zero-range boundary conditions. Scattering matrix elements for boson-dimer elastic scattering, breakup of a dimer into three particles and the time-reversed recombination process are derived. For vanishing total energy E, these quantities are given as closed-form functions of the two-body s -wave scattering length a and a three-body renormalization constant R0. The exact results obtained by this method are compared with those obtained using other methods. Differences in the functional dependence on R0 of the order of 2% are noted. Comparison with the hidden-crossing theory finds similar agreement with the functional dependence upon a and R0.