  The coupled equations for the scalar modes of the linearized Einstein
equations around Schwarzschild's spacetime were reduced by Zerilli to a 1+1
wave equation with a potential $V$, on a field $\Psi_z$. For smooth metric
perturbations $\Psi_z$ is singular at $r_s=-6M/(\ell-1)(\ell+2)$, $\ell$ the
mode harmonic number, and $V$ has a second order pole at $r_s$. This is
irrelevant to the black hole exterior stability problem, where $r>2M>0$, and
$r_s <0$, but it introduces a non trivial problem in the naked singular case
where $M<0$, then $r_s>0$, and the singularity appears in the relevant range of
$r$. We solve this problem by developing a new approach to the evolution of the
even mode, based on a {\em new gauge invariant function}, $\hat \Psi$ -related
to $\Psi_z$ by an intertwiner operator- that is a regular function of the
metric perturbation {\em for any value of $M$}. This allows to address the
issue of evolution of gravitational perturbations in this non globally
hyperbolic background, and to complete the proof of the linear instability of
the Schwarzschild naked singularity, by showing that a previously found
unstable mode is excitable by generic initial data. This is further illustrated
by numerically solving the linearized equations for suitably chosen initial
data.

General Relativity and Quantum Cosmology