  We consider a class of nonlinear Schroedinger equation in three space
dimensions with an attractive potential. The nonlinearity is local but rather
general encompassing for the first time both subcritical and supercritical (in
$L^2$) nonlinearities. We study the asymptotic stability of the nonlinear bound
states, i.e. periodic in time localized in space solutions. Our result shows
that all solutions with small initial data, converge to a nonlinear bound
state. Therefore, the nonlinear bound states are asymptotically stable. The
proof hinges on dispersive estimates that we obtain for the time dependent,
Hamiltonian, linearized dynamics around a careful chosen one parameter family
of bound states that "shadows" the nonlinear evolution of the system. Due to
the generality of the methods we develop we expect them to extend to the case
of perturbations of large bound states and to other nonlinear dispersive wave
type equations.
