Continuum singularities of a mean-field theory of collisions

Abstract

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)|Ψ〉=|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)|φi〉=|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)|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.