  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).
