Two interacting fermions in a one-dimensional harmonic trap: Matching the Bethe ansatz and variational approaches

In this work, combining the Bethe ansatz approach with the variational principle, we calculate the ground-state energy of the relative motion of a system of two fermions with spin up and spin down interacting via a delta-function potential in a one-dimensional (1D) harmonic trap. Our results show good agreement with the analytical solution of the problem, and provide a starting point for the investigation of more complex few-body systems where no exact theoretical solution is available.