Global unique continuation from a half space for the Schrödinger equation

Abstract

We obtain a global unique continuation result for the differential inequality $|(i{\partial }_t+\Delta )u|\le |V(x)u|$ in ${\mathbb{R}}^{n+1}$. This is the first result on global unique continuation for the Schrödinger 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.

arXiv commentRevised version, to appear in Journal of Functional Analysis