Dispersive estimates for the semi-classical Schr\"odinger equation in a strictly convex domain

We consider a model case for a strictly convex domain $\Omega\subset\mathbb{R}^d$ of dimension $d\geq 2$ with smooth boundary $\partial\Omega\neq\emptyset$ and we describe dispersion for the semi-classical Schr{\"o}dinger equation with Dirichlet boundary condition. More specifically, we obtain the optimal fixed time decay rate for the linear semi-classical flow : a loss of $(\frac ht)^{1/4}$ occurs with respect to the boundary less case due to repeated swallowtail type singularities. The result is optimal and implies corresponding Strichartz estimates.