Quantitative unique continuation for Schr\"odinger operators

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 $\Delta u + V u = 0$ 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 $\Delta u + V u = 0$. 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 $\Delta u + W \cdot \nabla u + V u = 0$.