Uniform asymptotics of eigenfunctions for the three-body Schrödinger operator in one-dimensional case.(English)Zbl 1180.81132

Møller, Jacob S. (ed.) et al., Quantum few-body systems. Proceedings of the joint physics/mathematics workshop on quantum few-body systems, Aarhus, Denmark, 19–20 March 2007. Melville, NY: American Institute of Physics (AIP) (ISBN 978-0-7354-0517-2/hbk). AIP Conference Proceedings 998, 101-112 (2008).
Summary: The three-body scattering problem with finite pair potentials for one-dimensional case is investigated. The asymptotic function $$\chi_0$$, which satisfies the three-body Schrödinger equation in whole configuration space outside of compact domain $$\Omega$$, where the supports of all three pair potentials cross each other, has been presented in a mathematically rigorous way. For large distances $$|x|\to\infty$$ the function $$\chi_0$$ determines the asymptotics of the solution up to the circular wave with smooth coefficient in whole configuration space. The method is based on analogies between few-body scattering problem and diffraction one of the plane wave on the system of half-transparent infinite screens. Presented here formalism are believed to be useful also for the few-body scattering problem of higher dimensions.
For the entire collection see [Zbl 1175.81002].

MSC:

 81U10 $$n$$-body potential quantum scattering theory 81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics 70F07 Three-body problems