Optimal rates of decay at infinity for solutions to Schrödinger equations

We prove rates of decay at infinity for solutions to variable-coefficient Schrödinger equations of the form $-\text{div}(A\mathrm{\nabla }u)+W\cdot \mathrm{\nabla }u+Vu=\lambda u$ 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 $\mathrm{\mid }\mathrm{\nabla }A\mathrm{\mid }$ at infinity. In particular, we prove a range of quantitative unique continuation-type results at infinity when $\mathrm{\mid }\mathrm{\nabla }A(\theta ,x)\mathrm{\mid }\le C(1+\mathrm{\mid }x\mathrm{\mid }{)}^{-\tau }$ 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 $\tau$.