  We consider a class of nonlinear Schrodinger equation in four and five space
dimensions with an attractive potential. The nonlinearity is local but rather
general encompassing for the first time both subcritical and supercritical (in
L2) nonlinearities. We show that the center manifold formed by localized in
space periodic in time solutions (bound states) is an attractor for all
solutions with a small initial data. The proof hinges on dispersive estimates
that we obtain for the time dependent, Hamiltonian, linearized dynamics around
a one parameter family of bound states that "shadows" the nonlinear evolution
of the system. The methods we employ are an extension to higher dimensions,
hence different linear dispersive behavior, and to rougher nonlinearities of
our previous results [10, 11, 7].
