  In this note, we show the existence of a special solution $u$ to defocusing
cubic NLS in $3d$, which lives in $H^{s}$ for all $s>0$, but scatters to a
linear solution in a very slow way. We prove for this $u$, for all
$\epsilon>0$, one has
$\sup_{t>0}t^{\epsilon}\|u(t)-e^{it\Delta}u^{+}\|_{\dot{H}^{1/2}}=\infty$. Note
that such a slow asymptotic convergence is impossible if one further pose the
initial data of $u(0)$ be in $L^{1}$. We expect that similar construction hold
the for other NLS models. It can been seen the slow convergence is caused by
the fact that there are delayed backward scattering profile in the initial
data, we also illustrate why $L^{1}$ condition of initial data will get rid of
this phenomena.
