Continuum Singularities of a Mean Field Theory of Collisions

Details Continuum Singularities of a Mean Field Theory of Collisions Continuum Singularities of a Mean Field Theory of Collisions

2005-04-08

arxiv.org/abs/math-ph/0504031v1

J.Math.Phys. 45 (2004) 1310

Physics

Mathematical Physics

21 pages, 14 figures

Scientific paper

10.1063/1.1666978

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.

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