  We consider the real time dynamics of $N$ noninteracting fermions in $d=1$.
They evolve in a trapping potential $V(x)$, starting from the equilibrium state
in a potential $V_0(x)$. We study the time evolution of the Wigner function
$W(x,p,t)$ in the phase space $(x,p)$, and the associated kernel which encodes
all correlation functions. At $t=0$ the Wigner function for large $N$ is
uniform in phase space inside the Fermi volume, and vanishes at the Fermi surf
over a scale $e_N$ being described by a universal scaling function related to
the Airy function. We obtain exact solutions for the Wigner function, the
density, and the correlations in the case of harmonic and inverse square
potentials, for several $V_0(x)$. In the large $N$ limit, near the edges where
the density vanishes, we obtain limiting kernels (of the Airy or Bessel types)
that retain the form found in equilibrium, up to a time dependent rescaling.
For non-harmonic traps the evolution of the Fermi volume is more complex.
Nevertheless we show that, for intermediate times, the Fermi surf is still
described by the same equilibrium scaling function, with a non-trivial time and
space dependent width which we compute analytically. We discuss the multi-time
correlations and obtain their explicit scaling forms valid near the edge for
the harmonic oscillator. Finally, we address the large time limit where
relaxation to the Generalized Gibbs Ensemble (GGE) was found to occur in the
"classical" regime $\hbar \sim 1/N$. Using the diagonal ensemble we compute the
Wigner function in the quantum case (large $N$, fixed $\hbar$) and show that it
agrees with the GGE. We also obtain the higher order (non-local) correlations
in the diagonal ensemble.
