Optimal rates of decay at infinity for solutions to Schrödinger equations

We prove rates of decay at infinity for solutions to variable-coefficient Schr\"{o}dinger equations of the form $-\text{div}(A \nabla u) + W \cdot \nabla u + V u = \lambda u$ in cylinders, $\mathbb{T}^d \times \mathbb{R}^m$. We assume that $W$ and $V$ are bounded and that $\lambda \in \mathbb{C}$. Our rates depend on the decay of $|\nabla A|$ at infinity. In particular, we prove a range of quantitative unique continuation-type results at infinity when $|\nabla A(\theta, x)| \le C (1 + |x|)^{-\tau}$ for $\tau \in [0,1]$. By adapting the methods in [KLP25], we construct explicit solutions to demonstrate the sharpness of our estimates for each such $\tau$.