Global unique continuation from a half space for the Schr\"odinger equation

We obtain a global unique continuation result for the differential inequality $|(i\partial_t+\Delta)u|\leq|V(x)u|$ in $\mathbb{R}^{n+1}$. This is the first result on global unique continuation for the Schr\"odinger equation with time-independent potentials $V(x)$ in $\mathbb{R}^{n}$. Our method is based on a new type of Carleman estimates for the operator $i\partial_t+\Delta$ on $\mathbb{R}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.