Dispersive Estimates for Nonlinear Schr\"odinger Equations with External Potentials

We consider the long time dynamics of nonlinear Schr"odinger equations with an external potential. More precisely, we look at Hartree type equations in three or higher dimensions with small initial data. We prove an optimal decay estimate, which is comparable to the decay of free solutions. Our proof relies on good control on a high Sobolev norm of the solution to estimate the terms in Duhamel’s formula.