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

Buslaev, V. S.; Levin, S. B.

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

Keywords:

three-body scattering problem; diffraction problem; uniform asymptotics