Exact solution for three particles interacting via zero-range potentials

J. H. MACEK, S. YU. OVCHINNIKOV AND G. GASANEO

PHYSICAL REVIEW A - ATOMIC, MOLECULAR AND OPTICAL PHYSICS

APS

Año: 2006 vol. 73 p. 1 - 14

1050-2947

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 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 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 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 method are compared with those obtained using other methods. Differences in the functional dependence on R0 -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. a and R0. 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 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 method are compared with those obtained using other methods. Differences in the functional dependence on R0 -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. a and R0.