Global well-posedness and scattering for a class of nonlinear Schröodinger equations below the energy space

We prove global well posedness and scattering for the nonlinear Schröodinger equation with power-type nonlinearity (mathematical formular) below the energy space, i.e., for s<1. In [15], J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao established polynomial growth of the Hs/x-norm of the solution, and hence global well posedness for initial data in Hs/x, provided s is sufficiently close to 1. However, their bounds are insufficient to yield scattering. In this paper, we use the a priori interaction Morawetz inequality to show that scattering holds in H^s(R^n) whenever s is larger than some value 0<s0(n,p)<1.