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

TL;DR

This paper establishes the first global unique continuation result for the Schrödinger equation with time-independent potentials in the entire space, using novel Carleman estimates, and extends the findings to certain parabolic equations.

Contribution

Introduces a new approach with Carleman estimates to prove global unique continuation for Schrödinger equations with time-independent potentials in all of bRb7n, a novel achievement.

Findings

First global unique continuation result for Schrödinger with time-independent potentials in bRb7n

Development of new Carleman estimates for the operator ipartial_t+Delta

Extension of the result to some parabolic equations

Abstract

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.