Decay rates at infinity for solutions to periodic Schrödinger 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 Schrödinger operators. The equation $\Delta u=Vu$ is studied as part of a broader class of elliptic evolution equations.

arXiv comment14 pages; second version includes extensions to the main results and an additional general result