Two interacting particles on the half-line: resolvent and scattering properties

Sebastian Egger, Joachim Kerner

Published 2017-02-02Version 1

In this paper we construct the resolvent and analyze scattering properties of the quantum two-body problem introduced in [arXiv:1504.08283] which describes two (distinguishable) particles moving on the half-line under the influence of singular two-particle interactions. Most importantly, due to the spatial localization of the interactions the two-body problem is of a non-separable nature. We will discuss the presence of embedded eigenvalues and using the detailed knowledge about the kernel of the resolvent we prove a version of the limiting absorption principle. Furthermore, by an appropriate adaptation of the Lippmann-Schwinger approach we are able to construct generalized eigenfunctions which consequently allow us to establish an explicit expression for the (on-shell) scattering amplitude. An approximation of the scattering amplitude in the weak-coupling limit is also derived.