On weak interaction between a ground state and a non-trapping potential

We show that ground states of the NLS moving at nonzero speed are asymptotically stable if they either stay far from the potential, or the potential is small, or the ground state has large speed. We search an effective Hamiltonian using the Birkhoff normal forms argument in [11], treating the potential as a perturbation. The so-called Fermi Golden Rule, which is used to describe the decay to 0 of the internal discrete modes of the ground state, is similar to that in [12]. The continuous modes dispersion requires the theory in [40] on charge transfer models.