  For the Hamiltonian operator H = -{\Delta}+V(x) of the Schr\"odinger Equation
with a repulsive potential, the problem of local decay is considered. It is
analyzed by a direct method, based on a new, L^2 bounded, propagation
observable. The resulting decay estimate, is in certain cases monotonic in
time, with no "Quantum Corrections". This method is then applied to some
examples in one and higher dimensions. In particular the case of the Wave
Equation on a Schwarzschild manifold is redone: Local decay, stronger than the
known ones are proved (minimal loss of angular derivatives and lower order of
radial derivatives of initial data). The method developed here can be an
alternative in some cases to the Morawetz type estimates, with L^2-multipliers
replacing the first order operators. It provides an alternative to Mourre's
method, by including thresholds and high energies.
