Smoothing effect for Schr\"odinger boundary value problems

We show the necessity of the non trapping condition for the plain smoothing effect ($H^{1\mathrm{/}2}$) for Schr"odinger equation with Dirichlet boundary conditions in exterior problems. We also give a class of trapped obstacles (Ikawa’s example) for which we can prove a weak ($H^{1\mathrm{/}2-ϵ}$) smoothing effect.