Continuum singularities of a mean-field theory of collisions

Consider a complex energy z for an N-particle Hamiltonian H and let χ be any wave packet accounting for any channel flux. The time-independent mean-field (TIMF) approximation of the inhomogeneous, linear equation (z-H) vertical bar Ψ>= vertical bar χ> consists of replacing Ψ by a product or Slater determinant φ of single-particle states φi. This results, under the Schwinger variational principle, in self-consistent TIMF equations (ηi-hi) vertical bar φi>= vertical bar χi> in single-particle space. The method is a generalization of the Hartree-Fock (HF) replacement of the N-body homogeneous linear equation (E-H) vertical bar Ψ>=0 by single-particle HF diagonalizations (ei-hi)|φi>=0. We show how, despite strong nonlinearities in this mean-field method, threshold singularities of the inhomogeneous TIMF equations are linked to solutions of the homogeneous HF equations