We consider the NLS on Schwarzschild manifold.For radial solutions with sufficiently localized initial data,global existence,L^p estimates and asymptotic completeness of the wave operators is proved

We consider the nonlinear Schrodinger equation with a pure power repulsive nonlinearity on Schwarzschild manifolds. Equations of this type arise when a nonlinear wave equation on a Schwarzschild manifold is written in Hamiltonian form, cf. [2], [10]. For radial solutions with sufficiently localized initial data, we obtain global existence, L-p estimates, and the existence and asymptotic completeness of the wave operators. Our approach is based on a dilation identity and global space-time estimates.