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Physical Review A

Stability of mean-field solutions for collisions

S. Kessal, BERTRAND G. GIRAUD

Published May 1, 1989• 0 citations
Publisher

Abstract

The time-independent mean-field theory of collisions converts an intractable linear problem, namely, the calculation of the many-body propagator (E-H${)}^{\mathrm{\ensuremath{-}}1}$, into a numerically solvable problem, namely, the coupled calculation of single-particle propagators (\ensuremath{\eta}-h${)}^{\mathrm{\ensuremath{-}}1}$. The latter problem, however, reflects its nonlinearity through many solutions, out of which only one is nonspurious and gives a good approximation to the unique solution (off shell) of the former, linear problem. We prove the stability of this nonspurious solution in the special case of a four-body, one-dimensional collision. The experience thus gained is used to generate transition-shell models in the case of a realistic, three-dimensional, four-body elastic collision.

DOI
10.1103/physreva.39.5014

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The time-independent mean-field theory of collisions converts an intractable linear problem, namely, the calculation of the many-body propagator (E-H)\ensuremath−1{)}^{\mathrm{\ensuremath{-}}1})\ensuremath−1, into a numerically solvable problem, namely, the coupled calculation of single-particle propagators (\ensuremath{\eta}-h)\ensuremath−1{)}^{\mathrm{\ensuremath{-}}1})\ensuremath−1. The latter problem, however, reflects its nonlinearity through many solutions, out of which only one is nonspurious and gives a good approximation to the unique solution (off shell) of the former, linear problem. We prove the stability of this nonspurious solution in the special case of a four-body, one-dimensional collision. The experience thus gained is used to generate transition-shell models in the case of a realistic, three-dimensional, four-body elastic collision.
The exact study of many-body microscopic systems is impossible when the number of particles is large (N ≥ 3). Approximate methods are then used. The time-independent mean-field (TIMF) approximation has been proposed for the description of collisions in many-body systems. Collision amplitudes are derived by the use of a variational principle and the choice of trial functions as products of single-particle orbitals. Resulting mean-field equations with a nonvanishing right-hand side turn out to be a generalization of the traditional Hartree or Hatree–Fock type equations. These TIMF equations are successfully solved numerically for the case of short-range forces. In this paper, we test the validity of this theory for the Coulomb interaction between two particles, that is, a long-range interaction. A numerical comparison between the exact and the mean-field solutions is conducted PACS Nos.: 31.15.Ne, 31.15.Pf, 21.45.+v,25.10.-i
A novel method is presented for implementation of the extended mean-field theory incorporating two-body collisions. At a given time, stochastic imaginary time propagation of occupied states are used to generate a convenient basis. The quantal collision terms, including memory effects, is then computed by a backward mean-field propagation of these single-particle states. The method is illustrated in an exactly solvable model. Whereas the usual TDHF fails to reproduce the long time evolution, a good agreement is found between the extended TDHF and the exact solution. PACS: 05.30.-d, 24.10.Cn, 05.60.+w
Abstract We consider quantum scattering of particles in media exhibiting strong dispersion degeneracy. In particular, we study flat-banded lattices and linearly dispersed energy bands. The former constitute a prime example of single-particle frustration while the latter show degeneracy at the few- and many-particle level. We investigate both impurity and two-body scattering and show that, quite generally, scattering does not occur, which we relate to the fact that transition matrices vanish on the energy shell. We prove that scattering is instead replaced by projections onto band-projected eigenstates of the interaction potential. We then use the general results to obtain localized flat band states that are eigenstates of impurity potentials with vanishing eigenvalues in one-dimensional flat bands and study the particular case of a sawtooth lattice. We also uncover the relation between certain solutions of one-dimensional systems that have been categorized as ‘strange’, and the scattering states in linearly dispersed continuum systems.