  We study the Cauchy problem for NLS with a class of $H^s$-super-critical data
\begin{align} & {\rm i}u_t +\Delta u+ \lambda |u|^{2\kappa} u =0, \quad
u(0)=u_0 \label{NLSabstract} \end{align} and show that \eqref{NLSabstract} is
globally well-posed and scattering in $\alpha$-modulation spaces
$M^{s,\alpha}_{2,1}$ ($\alpha\in [0,1), \ s> d\alpha/2-\alpha/\kappa$,
$\kappa\in \mathbb{N}$ and $\kappa \geq 2/d$) for the sufficiently small data.
Moreover, NLS is ill-posed in $M^{s,\alpha}_{2,1}$ if $s<
d\alpha/2-\alpha/\kappa$. In particular, we obtain a class of initial data
$u_0$ satisfying for any $M\gg 1$, \begin{align} \|u_0\|_2 \sim M^{1/\kappa-d/2
}, \ \ \|u_0\|_\infty \ =\infty , \ \ \|u_0\|_{M^{s,\alpha}_{2,1}} \geq
M^{(1-\alpha)/\kappa}, \ \ \ \|u_0\|_{B^{s(\kappa)}_{2,\infty}} =\infty
\nonumber \end{align} such that NLS is globally well-posed in
$M^{s,\alpha}_{2,1}$ if $\kappa>2/d, \ \alpha\in [0,1)\ d\alpha/2-\alpha/\kappa
<s < s(\kappa):= d/2-1/\kappa$. Such a kind of data are super-critical in
$H^{s(\kappa)}$ and have infinite amplitude.
