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Interacting one-dimensional quantum systems play a pivotal role in physics.Exact solutions can be obtained for the homogeneous case using the Bethe ansatzand bosonisation techniques. However, these approaches are not applicable whenexternal confinement is present. Recent theoretical advances beyond the Betheansatz and bosonisation allow us to predict the behaviour of one-dimensionalconfined systems with strong short-range interactions, and new experiments withcold atomic Fermi gases have already confirmed these theories. Here wedemonstrate that a simple linear combination of the strongly interactingsolution with the well-known solution in the limit of vanishing interactionsprovides a simple and accurate description of the system for all values of theinteraction strength. This indicates that one can indeed capture the physics ofconfined one-dimensional systems by knowledge of the limits using wavefunctions that are much easier to handle than the output of typical numericalapproaches. We demonstrate our scheme for experimentally relevant systems withup to six particles. Moreover, we show that our method works also in the caseof mixed systems of particles with different masses. This is an importantfeature because these systems are known to be non-integrable and thus notsolvable by the Bethe ansatz technique.

An interpolatory ansatz captures the physics of one-dimensional confined Fermi systems