Continuum Singularities of a Mean Field Theory of Collisions

Abstract

Consider a complex energy $z$ for a $N$-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 in replacing $Ψ$ by a product or Slater determinant $ϕ$ of single particle states $ϕ_i.$ This results, under the Schwinger variational principle, into self consistent TIMF equations $(η_i-h_i)|ϕ_i>=|χ_i>$ 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 $(e_i-h_i)|ϕ_i>=0.$ We show how, despite strong nonlinearities in this mean field method, threshold singularities of the {\it inhomogeneous} TIMF equations are linked to solutions of the {\it homogeneous} HF equations.