Decay rates at infinity for solutions to periodic Schr\"{o}dinger equations

Abstract

We consider the equation $\Delta u=Vu$ in exterior domains in $\mathbb{R}^2$ and $\mathbb{R}^3$, where $V$ has certain periodicity properties. In particular we show that such equations cannot have non-trivial superexponentially decaying solutions. As an application this leads to a new proof for the absolute continuity of the spectrum of particular periodic Schr\"{o}dinger operators. The equation $\Delta u=Vu$ is studied as part of a broader class of elliptic evolution equations.