  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.
