Optimal rates of decay at infinity for solutions to Schrödinger equations

Abstract

We prove rates of decay at infinity for solutions to variable-coefficient Schrödinger equations of the form $-\text{div}(A\nabla u)+W\cdot \nabla u+Vu=\lambda u$ 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 $|\nabla A|$ 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 $\tau$.

44 pages