  We study a weak interaction quench in a three-dimensional Fermi gas. We first
show that, under some general assumptions on time-dependent perturbation
theory, the perturbative expansion of the long-wavelength structure factor
$S(\bq)$ is not compatible with the hypothesis that steady-state averages
correspond to thermal ones. In particular, $S(\bq)$ does develop an analytical
component $\sim const. + O(q^2)$ at $\bq\to\bnot$, as implied by
thermalization, but, in contrast, it maintains a non-analytic part $\sim |\bq|$
characteristic of a Fermi-liquid at zero-temperature. In real space, this
non-analyticity corresponds to persisting power-law decaying density-density
correlations, whereas thermalization would predict only an exponential decay.
We next consider the case of a dilute gas, where one can obtain
non-perturbative results in the interaction strength but at lowest order in the
density. We find that in the steady-state the momentum distribution jump at the
Fermi surface remains finite, though smaller than in equilibrium, up to second
order in $k_F f_0$, where $f_0$ is the scattering length of two particles in
the vacuum. Both results question the emergence of a finite length scale in the
quench-dynamics as expected by thermalization.
