ON SCATTERING FOR NLS: FROM EUCLIDEAN TO HYPERBOLIC SPACE

We prove asymptotic completeness in the energy space for the nonlinear Schrodinger equation posed on hyperbolic space H-n in the radial case, for n >= 4, and 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 sort 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.