Scattering solutions to nonlinear Schr\"odinger equation with a long range potential

In this paper, we consider a nonlinear Schr\"odinger equation with a repulsive inverse-power potential. It is known that the corresponding stationary problem has a "radial" ground state. Here, the "radial" ground state is a least energy solution among radial solutions to the stationary problem. We prove that if radial initial data below the "radial" ground state has positive virial functional, then the corresponding solution to the nonlinear Schr\"odinger equation scatters. In particular, we can treat not only short range potentials but also long range potentials.