  We prove asymptotic completeness in the energy space for the nonlinear
Schrodinger equation posed on hyperbolic space in the radial case, in space
dimension at least 4, and for any energy-subcritical, defocusing, power
nonlinearity. The proof is based on simple Morawetz estimates and weighted
Strichartz estimates. We investigate the same question on spaces which kind of
interpolate between Euclidean space and hyperbolic space, showing that the
family of short range nonlinearities becomes larger and larger as the space
approaches the hyperbolic space. Finally, we describe the large time behavior
of radial solutions to the free dynamics.
