  We study the asymptotic behavior of solutions to the defocusing
mass-subcritical Hartree NLS $iu_t + \Delta u = F(u) = (|x|^{-\gamma}*|u|^2)u$
on $\mathbb{R}^d$, $d\geq 2$, $\frac{4}{3} < \gamma < 2$. We show that the
scattering problem associated to this equation is analytically well-posed in
the weighted spaces $\Sigma = H^1\cap\mathcal{F}H^1$ and $\mathcal{F}H^1$.
Furthermore, we show that the same problem fails to be analytically well-posed
for data in $L^2$. This constitutes an infinite loss of regularity between the
scattering problems in weighted spaces and in $L^2$. This further develops an
earlier investigation initiated by the author in which a finite breakdown of
regularity was proved for the $L^2$ scattering problem for the mass-subcritical
NLS with power nonlinearity $F(u) = |u|^pu$.
