Global Strichartz estimates for the Schr\"odinger equation with non zero boundary conditions and applications

We consider the Schr"odinger equation on a half space in any dimension with a class of nonhomogeneous boundary conditions including Dirichlet, Neuman and the so-called transparent boundary conditions. Building upon recent local in time Strichartz estimates (for Dirichlet boundary conditions), we obtain global Strichartz estimates for initial data in $H^s,0\le s\le 2$ and boundary data in a natural space ${\mathcal{H}}^s$. For $s\ge 1\mathrm{/}2$, the issue of compatibility conditions requires a thorough analysis of the ${\mathcal{H}}^s$ space. As an application we solve nonlinear Schr"odinger equations and construct global asymptotically linear solutions for small data. A discussion is included on the appropriate notion of scattering in this framework, and the optimality of the ${\mathcal{H}}^s$ space.