Exact Analysis of Delta-Function Attractive Fermions and Repulsive   Bosons in One-Dimension
  The Gaudin integral equation for the ground state of a one-dimensional delta-function attractive spin-1/2 fermions is solved in the form of power series. The first few terms of the asymptotic expansions for both strong and weak coupling cases are calculated analytically. The physical quantities such as the ground state energy are expressed in terms of a single dimensionless parameter gamma =c/D, where c is the coupling constant and D is the number density. The results agree with those obtained from the perturbation calculations, which include the one in the classical electrostatics originally by Kirchhoff. In the strong coupling limit, the connection to the solutions of the Lieb-Liniger integral equation for the ground state of a one-dimensional delta-function repulsive bose gas is shown explicitly. 
Toshiaki Iida and Miki Wadati