# Scattering properties of two singularly interacting particles on the half-line

## TL;DR

This paper investigates the scattering behavior of two distinguishable particles on the half-line with singular interactions, revealing non-separable dynamics, embedded eigenvalues, and explicit scattering amplitudes, including weak-coupling approximations.

## Contribution

It provides a detailed analysis of two-particle scattering with singular interactions, including the construction of generalized eigenfunctions and explicit scattering amplitude formulas.

## Key findings

- Identification of embedded eigenvalues
- Derivation of the limiting absorption principle
- Explicit expression for the scattering amplitude

## Abstract

We analyze scattering in a system of two (distinguishable) particles moving on the half-line $\overline{\rz}_+$ 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 obtained 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.