We consider the Schr\"odinger operator on a manifold with Euclidean
and/or hyperbolic ends, and discuss the maximal exponential decay
rates for generalized eigenfunctions. The proof depends on a variation
of the Mourre commutator argument. In the latter part of talk we
mention the asymptotic behaviors of $B^*$-eigenfucntions associated
with real eigenvalues above a critical energy, and also their relations
to the scattering matrix.
This talk is based on a joint work with Erik Skibsted, Aarhus University.