Stability for the time-dependent Hartree equation with positive energy

In this paper, we show that the H 1 solutions to the time-dependent Hartree equation i ∂ u ∂ t = 1 2 Δ u + 2 | x | u − ( | u | 2 ∗ 1 | x | ) u on R 3 are stable under the perturbation of parameters. We show that any sequence of H 1 solutions to the Hartree equations with perturbed parameters admits a weakly convergent subsequence, the weak limit of which is also an H 1 solution to the Hartree equation. To prove our stability result, we need to deal with the terms involving the potential 1 / | x | and new estimates will be employed.