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

We obtain a global unique continuation result for the differential inequality $\mathrm{\mid }(i{\mathrm{\partial }}_t+\mathrm{\Delta })u\mathrm{\mid }\le \mathrm{\mid }V(x)u\mathrm{\mid }$ 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 $i{\mathrm{\partial }}_t+\mathrm{\Delta }$ on ${\mathbb{R}}^{n+1}$. As a corollary of the result, we also obtain a new unique continuation result for some parabolic equations.