An analytically solvable three-body break-up model problem in hyperspherical coordinates

  • L. U. AncaraniEmail author
  • G. Gasaneo
  • D. M. Mitnik
Regular Article


An analytically solvable S-wave model for three particles break-up processes is presented. The scattering process is represented by a non-homogeneous Coulombic Schrödinger equation where the driven term is given by a Coulomb-like interaction multiplied by the product of a continuum wave function and a bound state in the particles coordinates. The closed form solution is derived in hyperspherical coordinates leading to an analytic expression for the associated scattering transition amplitude. The proposed scattering model contains most of the difficulties encountered in real three-body scattering problem, e.g., non-separability in the electrons’ spherical coordinates and Coulombic asymptotic behavior. Since the coordinates’ coupling is completely different, the model provides an alternative test to that given by the Temkin-Poet model. The knowledge of the analytic solution provides an interesting benchmark to test numerical methods dealing with the double continuum, in particular in the asymptotic regions. An hyperspherical Sturmian approach recently developed for three-body collisional problems is used to reproduce to high accuracy the analytical results. In addition to this, we generalized the model generating an approximate wave function possessing the correct radial asymptotic behavior corresponding to an S-wave three-body Coulomb problem. The model allows us to explore the typical structure of the solution of a three-body driven equation, to identify three regions (the driven, the Coulombic and the asymptotic), and to analyze how far one has to go to extract the transition amplitude.


Atomic and Molecular Collisions 


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Copyright information

© EDP Sciences, SIF, Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Laboratoire de Physique Moléculaire et des Collisions, UMR CNRS 7565Université de LorraineMetzFrance
  2. 2.Departamento de Física, Universidad Nacional del Sur and Consejo Nacional de Investigaciones Científicas y TécnicasBuenos AiresArgentina
  3. 3.Instituto de Astronomía y Física del Espacio (IAFE), Consejo Nacional de Investigaciones Científicas y Técnicas and FCEyNUniversidad de Buenos AiresBuenos AiresArgentina

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