Asymptotically linear solutions in H^1 of the 2-d defocusing nonlinear Schroedinger and Hartree equations

Abstract

In the 2-d setting, given an $H^1$ solution $v(t)$ to the linear Schr\"odinger equation $i\partial_t v +\Delta v =0$, we prove the existence (but not uniqueness) of an $H^1$ solution $u(t)$ to the defocusing nonlinear Schr\"odinger (NLS) equation $i\partial_t u + \Delta u -|u|^{p-1}u=0$ for nonlinear powers $2<p<3$ and the existence of an $H^1$ solution $u(t)$ to the defocusing Hartree equation $i\partial_t u + \Delta u -(|x|^{-\gamma}\star|u|^{2})u=0$ for interaction powers $1<\gamma<2$, such that $\|u(t)-v(t)\|_{H^1} \to 0$ as $t\to +\infty$. This is a partial result toward the existence of well-defined continuous wave operators $H^1 \to H^1$ for these equations. For NLS in 2-d, such wave operators are known to exist for $p\geq 3$, while for $p\leq 2$ it is known that they cannot exist. The Hartree equation in 2-d only makes sense for $0<\gamma<2$, and it was previously known that wave operators cannot exist for $0<\gamma\leq 1$, while no result was previously known in the range $1<\gamma<2$. Our proof in the case of NLS applies a new estimate of Colliander-Grillakis-Tzirakis (2008) to a strategy devised by Nakanishi (2001). For the Hartree equation, we prove a new correlation estimate following the method of Colliander-Grillakis-Tzirakis (2008).