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

• Published in: The European physical journal. D, Atomic, molecular, and optical physics Atomic, molecular, and optical physics, 2012-10, Vol.66 (10), Article 270
• Main Authors: Ancarani, L. U., Gasaneo, G., Mitnik, D. M.
• Format: Article
• Language: English
• Subjects: Applications of Nonlinear Dynamics and Chaos Theory; Atomic; Atomic and molecular collision processes and interactions; Atomic and molecular physics; Exact sciences and technology; General theories and models of atomic and molecular collisions and interactions (including statistical theories, transition state, stochastic and trajectory models, etc.); Molecular; Optical and Plasma Physics; Physical Chemistry; Physics; Physics and Astronomy; Quantum Information Technology; Quantum Physics; Regular Article; Spectroscopy/Spectrometry; Spintronics
• ISSN: 1434-6060; 1434-6079
• DOI: 10.1140/epjd/e2012-30353-4

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.