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

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.