We present a detailed analysis of the two-channel atom-molecule effective Hamiltonian for an ultracold\ud two-component homogeneous Fermi gas interacting near a Feshbach resonance. We particularly focus on the\ud two-body and many-body properties of the dressed molecules in such a gas. An exact result for the many-body\ud T matrix of the two-channel theory is derived by both considering coupled vertex equations and functionalintegral\ud methods. Making use of this result allows us to incorporate exactly into the many-body theory the\ud two-body physics of the Feshbach scattering by means of simple analytical formulas without any fitting\ud parameters. New interesting many-body effects are discussed in the case of narrow resonances. We give also a\ud description of the BEC-BCS crossover above and below TC. The effects of different approximations for the\ud self-energy of the dressed molecules are discussed. The single-channel results are derived as a special limit for\ud broad resonances. Moreover, through an analytic analysis of the BEC limit, the relation between the composite\ud boson of the single-channel model and the dressed-molecule of the two-channel model is established
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