Solutions of three-particle Faddeev equations above the breakup threshold via separable expansions of two-particle resolvents in a basis of two-particle pseudostates

Abstract

A separable expansion of the two-particle free resolvent in terms of two-particle pseudostates is used to convert Alt-Grassberger-Sandhas (AGS) integral equations into a set of effective two-body equations in spectator degrees of freedom. The resulting effective two-body equations are much like the multichannel Lippmann-Schwinger (LS) equations of inelastic scattering with real, energy independent, nonsingular potential matrices. Hence, they are more conducive to computations than the effective equations that ensue in the conventional approach based on separable expansions of two-particle transition operators. In particular, the problem of moving singularities of the conventional approach is avoided. The effective propagator matrix is complex and nondiagonal, and exhibits simple-pole singularities in diagonal elements corresponding to open rearrangement channels. These singularities can be handled by simple subtraction procedures well known from two-particle scattering. After regularization of the kernel, the set of coupled LS-type equations in the spectator momenta are solved rather straightforwardly via the Nystr & ouml;m method in which the integrals over spectator momenta are discretized using suitable quadrature rules. Solutions of effective two-body equations are then used to calculate the breakup amplitudes using the well-known relationship between rearrangement and breakup amplitudes. This proposed method has been tested on two models: (i) particle-dimer collisions in a three-boson model with s-wave separable pair potentials and (ii) an s-wave benchmark model with local pair potentials of then + d collisions. Calculations reported in the present article show that rather accurate results for elastic and breakup amplitudes can be obtained with pseudostates generated from a relatively small number of local basis functions in momentum space.