Global wellposedness and scattering for 3D energy critical Schrödinger equation with repulsive potential and radial data

In this paper, we show that the Cauchy problem of the 3D nonlinear Schrödinger equation with repulsive potential is globally wellposed if the initial data u0 is spherically symmetric and . We also prove that the scattering operator is holomorphic from the radial functions in Σ to themselves. In order to preclude the possible energy concentration, we first show the energy concentration may occur only at finite time by using the decay estimate of potential energy ∥u(t)∥6, then we preclude the possible finite time energy concentration by inductive arguments.