Absence of super-exponentially decaying eigenfunctions on Riemannian   manifolds with pinched negative curvature
  Let (X,g) be a metrically complete, simply connected Riemannian manifold with bounded geometry and pinched negative curvature, i.e. there are constants a>b>0 such that -a^2<K<-b^2 for all sectional curvatures K. Here bounded geometry is used in the sense that all covariant derivatives of the Riemannian curvature tensor are bounded and the injectivity radius is uniformly bounded below by a positive constant. We show that there are no superexponentially decaying eigenfunctions of the Laplacian of g. We also show the analogous conclusion for other geometric operators, and prove a theorem with the assumptions and conclusions localized near infinity. 
Andras Vasy and Jared Wunsch