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The coupled equations for the scalar modes of the linearized Einstein equations around Schwarzschilds spacetime were reduced by Zerilli to a (1+1) wave equation ∂2Psi z/∂t 2 + H z = 0, where H = −∂2/∂x2 + V (x) is the Zerilli Hamiltonian and x is the tortoise radial coordinate. From its definition, for smooth metric perturbations the field Psiz is singular at rs = −6M/( L−1)( L+2), with L being the mode harmonic number. The equation Psiz obeys is also singular, since V has a second-order pole at rs. This is irrelevant to the black hole exterior stability problem, where r > 2M > 0, and rs < 0, but it introduces a non-trivial problem in the naked singular case where M <0, then rs > 0, and the singularity appears in the relevant range of r (0 < r < ∞). We solve this problem by developing a new approach to the evolution of the even mode, based on a new gauge invariant function, Psih, that is a regular function of the metric perturbation for any value of M. The relation of Psih to Psiz is provided by an intertwiner operator. The spatial pieces of the (1+1) wave equations that Psih and Psiz obey are related as a supersymmetric pair of quantum Hamiltonians H and H . For M < 0, H has a regular potential and a unique self-adjoint extension in a domain D defined by a physically motivated boundary condition at r = 0. This allows us to address the issue of evolution of gravitational perturbations in this non-globally hyperbolic background. This formulation is used to complete the proof of the linear instability of the Schwarzschild naked singularity, by showing that a previously found unstable mode belongs to a complete basis of H in D, and thus is excitable by generic initial data. This is further illustrated by numerically solving the linearized equations for suitably chosen initial data