The single-particle energies for a one-dimensional system of interacting fermions via a repulsive \ensuremath{\delta}-function potential are self-consistently calculated using the Brueckner and Gammel method. The breakdown of the usual Fermi liquid picture is signaled in this approach by the occurrence of a negative gap at \ifmmode\pm\else\textpm\fi{}
kF between the lowest energy of a particle state (|k|>
kF) and the highest energy of a hole state (|k|
kF); a normal ground-state configuration with all the hole states double occupied is thus unstable. Two possible pictures getting account of the rearrangement of the particles in a stable configuration are discussed: (1) an effective potential simulating a normal Fermi liquid behavior; (2) a condensate of fermions and a reduced Fermi sea normally occupied. The ground-state energy computed within these schemes is in remarkable agreement with the Bethe-Ansatz results for all values of the model parameters.