  We consider two particular 1D quantum many-body systems with local
interactions related to the root system $C_N$. Both models describe identical
particles moving on the half-line with non-trivial boundary conditions at the
origin, and they are in many ways complementary to each other. We discuss the
Bethe Ansatz solution for the first model where the interaction potentials are
delta-functions, and we find that this provides an exact solution not only in
the boson case but even for the generalized model where the particles are
distinguishable. In the second model the particles have particular momentum
dependent interactions, and we find that it is non-trivial and exactly solvable
by Bethe Ansatz only in case the particles are fermions. This latter model has
a natural physical interpretation as the non-relativistic limit of the massive
Thirring model on the half-line. We establish a duality relation between the
bosonic delta-interaction model and the fermionic model with local momentum
dependent interactions. We also elaborate on the physical interpretation of
these models. In our discussion the Yang-Baxter relations and the Reflection
equation play a central role.
