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We describe the asymptotic behavior of small energy solutions of an NLS witha trapping potential. In particular we generalize work of Soffer and Weinstein,and of Tsai et. al. The novelty is that we allow generic spectra associated tothe potential. This is yet a new application of the idea to interpret thenonlinear Fermi Golden Rule as a consequence of the Hamiltonian structure.

On small energy stabilization in the NLS with a trapping potential