Adiabatic limit of the Faddeev equations

The adiabatic scattering of two heavy particles, interacting by the exchange of a third light particle, is considered from the Faddeev viewpoint. The Faddeev equations are reduced to an equivalent infinite set of coupled integral equations, which like the equations of the close-coupling scheme have two-body kinematics, and effective two-body interactions, but unlike the close-coupling equations, properly include the effects of breakup states. Complete sets of eigenstates of the kernels of the two-body Lippmann-Schwinger equations (Sturmian states) are used to effect the expansions. For the special case of separable interactions the equivalent set is shown to be finite. The effective interactions of the theory are examined in the adiabatic and static limits and compared with the potentials of the close-coupling scheme. In the limiting case the interactions are determined by a reduced form of the three-particle propagator and found to be local, leading to an alternative set of coupled differential equations for the three-body elastic amplitude. The effective potentials and their limiting forms are constructed for three specific examples, the Yamaguchi interaction, a nonrelativistic model of the nucleon-nucleon interaction, and a model of heavy-ion transfer scattering using delta-shell interactions