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Title: Fermi-liquid behavior of weakly interacting fermions above the pairing scale
Plain: 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: 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.
Settled by: A convergent multiscale (renormalization-group) proof of Fermi-liquid bounds in 3D down to an exponentially small temperature.
Kind: well-posed
Answer type: proof
References: (empty)
Literature status: open
Status note: 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.
Posed since: (empty)