  We investigate the energy spectrum of systems of two, three and four spin-1/2
fermions with short range attractive interactions both exactly, and within the
scattering length approximation. The formation of molecular bound states and
the ferromagnetic transition of the excited scattering state are examined
systematically as a function of the 2-body scattering length. Identification of
the upper branch (scattering states) is discussed and a general approach valid
for systems with many particles is given. We show that an adiabatic
ferromagnetic transition occurs, but at a critical transition point kF a much
higher than predicted from previous calculations, almost all of which use the
scattering length approximation. In the 4-particle system the discrepancy is a
factor of 2. The exact critical interaction strength calculated in the
4-particle system is consistent with that reported by experiment. To make
comparisons with the adiabatic transition, we study the quench dynamics of the
pairing instability using the eigenstate wavefunctions.
