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In this work, combining the Bethe ansatz approach with the variationalprinciple, we calculate the ground state energy of the relative motion of asystem of two fermions with spin up and down interacting via a delta-functionpotential in a 1D harmonic trap. Our results show good agreement with theanalytical solution of the problem, and provide a starting point for theinvestigation of more complex few-body systems where no exact theoreticalsolution is available.

Two interacting fermions in a 1D harmonic trap: matching the Bethe ansatz and variational approaches