  We consider the cubic nonlinear Schr\"{o}dinger equation in two space
dimensions with an attractive potential. 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, localized in space initial
data, converge to the set of bound states. Therefore, the center manifold in
this problem is a global attractor. The proof hinges on dispersive estimates
that we obtain for the non-autonomous, non-Hamiltonian, linearized dynamics
around the bound states.
