  We consider the wave and Schr{\"o}dinger equations with Dirichlet boundary
conditions in the exterior of a ball in $R^d$. In dimension $d = 3$ we
construct a sharp, global in time parametrix and then proceed to obtain sharp
dispersive estimates, matching the $R^3$ case, for all frequencies (low and
high). If $d $\ge$ 4$, we provide an explicit solution to the wave equation
localized at large frequency $1/h$ with data $\delta_{Q_0}$, where $Q_0$ is a
point at large distance s from the center of the ball : taking $s \sim
h^{-1/3}$, the decay rate of that solution exhibits a $(t/h)^{d-3}/4$ loss with
respect to the boundary less case, that occurs at $t \sim 2s$ with an
observation point being symmetric to $Q_0$ with respect to the center of the
ball (at the Poisson Arago spot). A similar counterexample is also obtained for
the Schr{\"o}dinger flow.
