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

TL;DR

This paper investigates the unique continuation property for solutions to fractional and higher-order Schrödinger equations with potentials in certain function classes, extending known results to a broader range of fractional orders.

Contribution

It establishes a unique continuation theorem for fractional Schrödinger operators with potentials in specific function classes, covering the full range of fractional orders between 1 and 2.

Findings

Unique continuation holds for fractional Schrödinger operators with potentials in certain L^p classes.

Results cover the full fractional order range 1<α<2 in two dimensions.

The paper extends previous unique continuation results to higher-order and fractional cases.

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.