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Regime of Validity of the Pairing Hamiltonian in the Study of Fermi Gases

The ground state energy and pairing gap of the interacting Fermi gases calculated by the {\it ab initio} stochastic method are compared with those estimated from the Bardeen-Cooper-Schrieffer pairing Hamiltonian. We discuss the ingredients of this Hamiltonian in various regimes of interaction strength. In the weakly interacting ($1/ak_F <<0$) regime the BCS Hamiltonian should describe Landau quasi-particle energies and interactions, on the other hand in the strongly pairing regime, that is $1/ak_F \gtrsim 0$, it becomes part of the bare Hamiltonian. However, the bare BCS Hamiltonian is not adequate for describing atomic gases in the regime of weak to moderate interaction strength $-\infty < 1/ak_F <0$ such as $ak_F \sim -1$. PACS: 05.30.Fk, 03.75.Ss, 21.65.+f