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We consider the wave and Schr\"odinger equations with Dirichlet boundaryconditions in the exterior of a ball in $\mathbb{R}^d$. In dimension $d = 3$ weconstruct a sharp, global in time parametrix and then proceed to obtain sharpdispersive estimates, matching the $\mathbb{R}^3$ case, for all frequencies(low and high). If $d \ge 4$, we provide an explicit solution to the waveequation 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 $(t/h)^{d-3}/4$ losswith respect to the boundary less case, that occurs at $t \sim 2s$ with anobservation point being symmetric to $Q_0$ with respect to the center of theball (at the Poisson Arago spot). A similar counterexample is also obtained forthe Schr\"odinger flow.

Dispersion for the wave and Schr\"odinger equations outside a ball and counterexamples