SMF

Radiation conditions and scattering theory for threeparticle Hamiltonians

Radiation conditions and scattering theory for threeparticle Hamiltonians

Dimitri YAFAEV
  • Année : 1992
  • Tome : 210
  • Format : Papier, Électronique
  • Langue de l'ouvrage :
    Anglais
  • Class. Math. : 35P25
  • Pages : 355-384
  • DOI : 10.24033/ast.197

The correct form of radiation conditions is found in scattering problem for threeparticle Hamiltonians $H$. For example, in a cone $\Gamma $ of the configuration space where all pair potentials are vanishing the radiation conditions-estimate has the following form. Let $\nabla ^{(s)}$, $\nabla ^{(s)}u(x) = \nabla u(x) - |x|^{-2} \langle \nabla u(x),x \rangle x,$ be the projection of the gradient $\nabla $ on the plane, orthogonal to $x$, and let $\xi $ be the characteristic function of $\Gamma $. Then the operator $\xi (|x| + 1)^{-1/2}\nabla ^{(s)}$ is locally (away from thresholds and eigenvalues of $H$) $H$-smooth (in the sense of T.Kato). In cones where some of pair potentials are not vanishing radiation conditions-estimates have similar (though weaker) form with the gradient replaced by its projection on a certain subspace. Such estimates allows us to give an elementary proof of the asymptotic completeness for three-particle systems in the framework of the theory of smooth perturbations.

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