Fermi-liquid behavior of weakly interacting fermions above the pairing scale
In plain words
Show that electrons with weak repulsion in a 3D continuum act like slightly modified free particles down to very low temperatures. At the lowest temperatures they are expected to pair, so the statement must stop above that scale.
Precise statement
Spin-$1/2$ fermions in $R^{3}$ with dispersion $k^{2}/(2m)$ and weak short-range repulsion $\lambda v$. Prove that for T above the pairing scale ($T \ge \exp(-c/\lambda^{2}) E_F$, say) the momentum distribution has a jump of size $Z = 1 - \operatorname{O}(\lambda^{2})$ smeared only on scale T, and $\operatorname{Im} \Sigma(\omega, k_F) \sim \omega^{2} + (\pi T)^{2}$. Answer: a proof with explicit constants.
What would settle it
A convergent multiscale (renormalization-group) proof of Fermi-liquid bounds in 3D down to an exponentially small temperature.
Status in the literature
In $2\mathrm{D}$, Disertori and Rivasseau (2000) controlled jellium above an exponentially small temperature, and Feldman, Knorrer and Trubowitz (2003-2004) proved a $T = 0$ Fermi liquid for Fermi surfaces without inversion symmetry.