Traces of integrability in scattering of one-dimensional dimers on a barrier

We consider molecules made of two one-dimensional short-range-interacting bosonic atoms. We show that in the process of scattering of these molecules off a narrow barrier, odd incident waves produce no unbound atoms, even when the incident energy exceeds the dissociation threshold. We demonstrate that such a feature originates from the fact that our system shares a part of its spectrum, namely its odd eigenstates, with a Bethe Ansatz integrable model, a delta-interaction Hamiltonian with local and non-local terms, associated with a reflection group C-2-a full symmetry of a square. This partial integrability induces prohibition of chemical reactions, allowing us to make predictions about scattering processes. We suggest several experimental implementations of the effect. We also propose to use the monomer production as an alternative read-out channel in an atom interferometer: unlike in the standard interferometric schemes, no spatial separation of the output channels will be required. Our approach offers an example of use of integrable problems beyond the realm of fully integrable models.