Simple approximation for the Bethe-ansatz solution of the one-dimensional Fermi gas

We present a simple approximation scheme for the solution of the integral equations resulting from the Bethe-ansatz diagonalization of the one-dimensional Fermi gas with {δ}-function attraction. These equations arise for the Hubbard model with attractive interaction in the limit of weak coupling and low density. We obtain the ground-state energy as a function of coupling and density in very good agreement with numerical solutions, as well as a value for the parameter determining the gap in the magnetic excitation spectrum which strongly supports a conjecture of Larkin and Sak.