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A generalized Lieb-Liniger model

In 1963, Lieb and Liniger solved exactly a one dimensional model of bosons interacting by a repulsive δ-potential and calculated the ground state in the thermodynamic limit. In the present work, we extend this model to a potential of three δ-functions, one of them is repulsive and the other two are attractive, modeling some aspects of the interaction between atoms, and present an approximate solution for a dilute gas. In this limit, for low energy states, the results are found to be reduced to the ones of an effective Lieb Liniger model with an effective δ-function of strength $c_{eff}$ and the regime of stability is identified. This may shed light on some aspects of interacting bosons.