  We investigate here the Cooper pairing of fermionic atoms with mismatched
fermi surfaces using a variational construct for the ground state. We determine
the state for different values of the mismatch of chemical potential for weak
as well as strong coupling regimes including the BCS BEC cross over region. We
consider Cooper pairing with both zero and finite net momentum. Within the
variational approximation for the ground state and comparing the thermodynamic
potentials, we show that (i) the LOFF phase is stable in the weak coupling
regime, (ii) the LOFF window is maximum on the BEC side near the Feshbach
resonance and (iii) the existence of stable gapless states with a single fermi
surface for negative average chemical potential on the BEC side of the Feshbach
resonance.
