  In the Luttinger model with non-local interaction we investigate, by exact
analytical methods, the time evolution of an inhomogeneous state with a
localized fermion added to the non interacting ground state. In absence of
interaction the averaged density has two peaks moving in opposite directions
with constant velocities. If the state is evolved with the interacting
Hamiltonian two main effects appear. The first is that the peaks have
velocities which are not constant but vary between a minimal and maximal value.
The second is that a dynamical `Landau quasi-particle weight' appears in the
oscillating part of the averaged density, asymptotically vanishing with time,
as consequence of the fact that fermions are not excitations of the interacting
Hamiltonian.
