  In this paper we provide an extension of the model discussed in
[arXiv:1504.08283] describing two singularly interacting particles on the
half-line. In this model, the particles are interacting only whenever at least
one particle is situated at the origin. Stimulated by [arXiv:1503.08814] we
then provide a generalisation of this model in order to include additional
interactions between the particles leading to a molecular-like state. We give a
precise mathematical formulation of the Hamiltonian of the system and perform
spectral analysis. In particular, we are interested in the effect of the
singular two-particle interactions onto the molecule.
