  In the case of compact quantum graphs, many-particle models with singular
two-particle interactions where introduced in [arXiv:1207.5648,
arXiv:1112.4751] to provide a paradigm for further studies on many-particle
quantum chaos. In this note, we discuss various aspects of such singular
interactions in a two-particle system restricted to the half-line
$\mathbb{R}_+$. Among others, we give a description of the spectrum of the
two-particle Hamiltonian and obtain upper bounds on the number of eigenstates
below the essential spectrum. We also specify conditions under which there is
exactly one such eigenstate. As a final result, it is shown that the ground
state is unique and decays exponentially as $\sqrt{x^2+y^2} \to \infty$.
