Asymptotic behavior of the eigenfunctions of the many-particle Schrödinger operator. I: One-dimensional particles.(English)Zbl 1160.81476

Suslina, T. (ed.) et al., Spectral theory of differential operators. M. Sh. Birman 80th anniversary collection. Providence, RI: American Mathematical Society (AMS) (ISBN 978-0-8218-4738-1/hbk). Translations. Series 2. American Mathematical Society 225; Advances in the Mathematical Sciences 62, 55-71 (2008).
Summary: We study the three-body nonrelativistic quantum scattering problem with rapidly decreasing at infinity pair potentials for the case of one-dimensional particles. We consider the solution $$\chi(z,q)$$ of the corresponding Schrödinger equation that can be interpreted as the scattered plane wave $$e^{i\langle z,q\rangle}$$ and find its asymptotic behavior as $$|z|\to\infty$$. More precisely, we explicitly construct a function $$\chi_0(z,q)$$ that determines the asymptotic behavior of the solution up to a diverging circle wave with a smooth amplitude coefficient. The method is based on analogies between the quantum scattering problem and the diffraction of the plane wave by a system of half-transparent infinite screens. We believe that the formalism can be useful also for studying many-dimensional particle scattering and for the case of long range pair potentials.
For the entire collection see [Zbl 1152.47002].

MSC:

 81U10 $$n$$-body potential quantum scattering theory 81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis