  We revisit a result by Cuccagna, Kirr and Pelinovsky about the cubic
nonlinear Schr\" odinger equation (NLS) with an attractive localized potential
and a time-dependent factor in the nonlinearity. We show that, under generic
hypotheses on the linearization at 0 of the equation, small energy solutions
are asymptotically free. This is yet a new application of the hamiltonian
structure, continuing a program initiated in a paper by Bambusi and Cuccagna.
