Exact solutions of two complementary one-dimensional quantum many-body systems on the half-line

We consider two particular 1D quantum many-body systems with localinteractions related to the root system $C_N$. Both models describe identicalparticles moving on the half-line with non-trivial boundary conditions at theorigin, and they are in many ways complementary to each other. We discuss theBethe Ansatz solution for the first model where the interaction potentials aredelta-functions, and we find that this provides an exact solution not only inthe boson case but even for the generalized model where the particles aredistinguishable. In the second model the particles have particular momentumdependent interactions, and we find that it is non-trivial and exactly solvableby Bethe Ansatz only in case the particles are fermions. This latter model hasa natural physical interpretation as the non-relativistic limit of the massiveThirring model on the half-line. We establish a duality relation between thebosonic delta-interaction model and the fermionic model with local momentumdependent interactions. We also elaborate on the physical interpretation ofthese models. In our discussion the Yang-Baxter relations and the Reflectionequation play a central role.Comment: 15 pages, a mistake corrected changing one of our conclusion