Scattering and localized states for defocusing nonlinear Schr\"odinger equations with potential

We study the large time behavior of global energy class solutions of the one dimensional nonlinear Schr\"odinger equation with a general localized potential term and a defocusing nonlinear term. By using a new type of interaction Morawetz estimate localized to an exterior region, we prove that these solutions decompose into a free wave and a weakly localized part which is asymptotically orthogonal to any fixed free wave. We further show that the L² norm of this weakly localized part is concentrated in the region |x| t^1/2+, and that the energy (Ḣ¹) norm is concentrated in |x| t^1/3+.