BCS theory tested in an exactly solvable fermion fluid

Remarkably the one-dimensional (1D) many-fermion fluid with pairwise-attractive \ensuremath{\delta}-function interactions is exactly solvable in that one can determine the exact many-body ground-state energy and chemical potential for all values of the coupling strength and/or density. Bardeen-Cooper-Schrieffer (BCS) theory is tested in this model by numerically determining the BCS total ground-state energy and chemical potential as a function of the coupling strength and/or density, and comparing with the exact results. As is the case for 2D and 3D theories, two regimes are apparent: (a) a BCS-proper regime of weakly coupled, overlapping Cooper pairs and (b) a Bose-gas regime of strongly interacting fermions which pair to form an ideal Bose gas at low density. In the two extremes the BCS energy and chemical potential are identical to the exact values and are moderately close for intermediate coupling and/or density.