Quantitative unique continuation for Schrödinger operators

Abstract

We investigate the quantitative unique continuation properties of solutions to second order elliptic equations with singular lower order terms. The main theorem presents a quantification of the strong unique continuation property for $\Delta +V$. That is, for any non-trivial $u$ that solves $\Delta u+Vu=0$ in some open, connected subset of ${\mathbb{R}}^n$, we estimate the vanishing order of solutions in terms of the $L^t$-norm of $V$. Our results apply to all $t>\frac{n}{2}$ and $n\ge 3$. With these maximal order of vanishing estimates, we employ a scaling argument to produce quantitative unique continuation at infinity estimates for global solutions to $\Delta u+Vu=0$. To handle $V\in L^t$ for every $t\in (\frac{n}{2},\infty ]$, we prove a novel $L^p-L^q$ Carleman estimate by interpolating a known $L^p-L^2$ estimate with a new endpoint Carleman estimate. This new Carleman estimate may also be used to establish improved order of vanishing estimates for equations with a first order term, those of the form $\Delta u+W\cdot \nabla u+Vu=0$.