On asymptotic stability in energy space of ground states of NLS in 1D

We transpose work by T. Mizumachi to prove smoothing estimates for dispersive solutions of the linearization at a ground state of a Nonlinear Schrödinger equation (NLS) in 1D. As an application we extend to dimension 1D a result on asymptotic stability of ground states of NLS proved by Cuccagna and Mizumachi for dimensions 3.