On unique continuation for Schr\"odinger operators of fractional and higher orders

Abstract

In this note we study the property of unique continuation for solutions of $|(-\Delta)^{\alpha/2}u|\leq|Vu|$, where $V$ is in a function class of potentials including $\bigcup_{p>n/\alpha}L^p(\mathbb{R}^n)$ for $n-1\leq\alpha<n$. In particular, when $n=2$, our result gives a unique continuation theorem for the fractional ($1<\alpha<2$) Schr\"odinger operator $(-\Delta)^{\alpha/2}+V(x)$ in the full range of $\alpha$ values.