We obtain fixed time decay rate for the linear (semi-classical) Schrödinger flow inside a general strictly convex domain. 
Corresponding Strichartz estimates allow to solve the cubic NLS in the natural energy class on such 3D convex domains. This is joint work with O. Ivanovici. We will start by reviewing earlier work (with O. Ivanovici, R. Lascar and G. Lebeau) on the wave equation on such domains, as the optimal decay rates in all regimes for waves turn out to be crucial to deal with Schrödinger.