3 Ground state of one - dimensional bosons with delta interaction : link to the BCS model

We have found that the Bethe roots describing the ground state of the integrable one-dimensional model of interacting bosons with weakly repulsive two-body delta interactions satisfy the set of equations appearing in the strong coupling limit of the integrable BCS model. The BCS model describes electron-electron interactions in weakly disordered metallic grains. It follows that the Bethe roots of the weakly interacting boson model are given by the zeros of Laguerre polynomials. We have thus shown a direct link, in the context of integrable one-dimensional models, between bosons interacting via weakly repulsive two-body delta-interactions and strongly interacting Cooper pairs.