  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.
