Asymptotically Linear Solutions in H of the 2-d Defocusing Nonlinear Schrödinger and Hartree Equations

In the 2-d setting, given an H solution v(t) to the linear Schrödinger equation i∂tv + ∆v = 0, we prove the existence (but not uniqueness) of an H solution u(t) to the defocusing nonlinear Schrödinger (NLS) equation i∂tu+∆u− |u|p−1u = 0 for nonlinear powers 2 < p < 3 and the existence of an H solution u(t) to the defocusing Hartree equation i∂tu +∆u − (|x|−γ ? |u|)u = 0 for interaction powers 1 < γ < 2, such that ‖u(t) − v(t)‖H1 → 0 as t → +∞. This is a partial result toward the existence of well-defined continuous wave operators H → H for these equations. For NLS in 2-d, such wave operators are known to exist for p ≥ 3, while for p ≤ 2 it is known that they cannot exist. The Hartree equation in 2-d only makes sense for 0 < γ < 2, and it was previously known that wave operators cannot exist for 0 < γ ≤ 1, while no result was previously known in the range 1 < γ < 2. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis [4] to a strategy devised by Nakanishi [15]. For the Hartree equation, we prove a new correlation estimate following the method of [4].