Continuum Singularities of a Mean Field Theory of Collisions

TL;DR

This paper explores how singularities in a mean field approximation for multi-particle collisions relate to solutions of simpler, homogeneous equations, revealing insights into complex quantum interactions.

Contribution

It demonstrates the connection between threshold singularities in inhomogeneous TIMF equations and solutions of homogeneous Hartree-Fock equations, extending understanding of nonlinear quantum systems.

Findings

Threshold singularities are linked to homogeneous HF solutions.

Nonlinearities in TIMF equations are analyzed.

Insights into complex collision processes are provided.

Abstract

Consider a complex energy $z$ for a $N$-particle Hamiltonian $H$ and let $\chi$ be any wave packet accounting for any channel flux. The time independent mean field (TIMF) approximation of the inhomogeneous, linear equation $(z-H)|\Psi>=|\chi>$ consists in replacing $\Psi$ by a product or Slater determinant $\phi$ of single particle states $\phi_i.$ This results, under the Schwinger variational principle, into self consistent TIMF equations $(\eta_i-h_i)|\phi_i>=|\chi_i>$ in single particle space. The method is a generalization of the Hartree-Fock (HF) replacement of the $N$-body homogeneous linear equation $(E-H)|\Psi>=0$ by single particle HF diagonalizations $(e_i-h_i)|\phi_i>=0.$ 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.