  In the present work, we start from a minimal Hamiltonian for Fermi systems
where the s-wave scattering is the only low energy constant at play. Many-Body
Perturbative approach that is usually valid at rather low density is first
discussed. We then use the resummation technique with the ladder approximation
to obtain compact expressions for both the energy and/or the on-shell
self-energy in infinite spin-degenerated systems. Diagrammatic resummation
technique has the advantage in general to be predictive in a region of density
larger compared to many-body perturbation theory. It also leads to
non-diverging limit as $|a_s| \rightarrow + \infty$. Still, the obtained
expressions are rather complex functional of the Fermi momentum $k_F$. We
introduce the full phase-space average or the partial phase-space methods
respectively applied to the energy or to the self-energy to simplify their
dependences in terms of $(a_s k_F)$ while keeping the correct limit at low
density and the non-diverging property at large $|a_s k_F|$. Quasi-particle
properties of Fermi system in various regime of density and scattering length
are then illustrated. Our conclusion is that such simplified expressions where
the direct link is made with the low energy constant without fine-tuning can
provide a clear guidance to obtain density functional theory beyond the
perturbative regime. However, quasi-particle properties close or near unitary
cannot be reproduced unless this limit is explicitly used as a constraint. We
finally discuss how such approximate treatment of quasi-particle can guide the
development of density functional theory for strongly interacting Fermi
systems.
