Asymptotic Dynamics of Nonlinear Schr\"odinger Equations with Many Bound States

We consider a nonlinear Schr\"odinger equation with a bounded local potential in $R^3$. The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order $n$ in $H^1$, and that its ground state component is larger than $n^{3-\epsilon}$ with $\epsilon>0$ small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.