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We investigate the scattering theory for the nonlinear Schr\"{o}dingerequation $i \partial_{t}u+ \Delta u+\lambda|u|^\alpha u=0$ in$\Sigma=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scatteringstates $u^{\pm}$ exist in $\Sigma$ when $\alpha_{d}<\alpha<\frac{4}{d-2}$,$d\geq3$, $\lambda\in \mathbb{R}$ with certain smallness assumption on theinitial data $u_{0}$, and when $\alpha(d)\leq \alpha< \frac{4}{d-2}$($\alpha\in[\alpha(d), \infty)$, if $d=1,2$), $\lambda>0$ under suitable conditions on$u_{0}$, where $\alpha_{d}$, $\alpha(d)$ are the positive root of thepolynomial $dx^{2}+dx-4$ and $dx^{2}+(d-2)x-4$ respectively. Specially, when$\lambda>0$, we obtain the existence of $u^{\pm}$ in $\Sigma$ for $u_{0}$ belowa mass-energy threshold$M[u_{0}]^{\sigma}E[u_{0}]<\lambda^{-2\tau}M[Q]^{\sigma}E[Q]$ and satisfying anmass-gradient bound $\|u_{0}\|_{L^{2}}^{\sigma}\|\nablau_{0}\|_{L^{2}}<\lambda^{-\tau}\|Q\|_{L^{2}}^{\sigma}\|\nabla Q\|_{L^{2}}$ with$\frac{4}{d}<\alpha<\frac{4}{d-2}$($\alpha\in (\frac{4}{d}, \infty)$, if$d=1,2$), and also for oscillating data at critical power $\alpha=\alpha(d)$,where $\sigma=\frac{4-(d-2)\alpha}{\alpha d-4}$, $\tau=\frac{2}{\alpha d-4}$and $Q$ is the ground state. We also study the convergence of $u(t)$ to thefree solution $e^{it\Delta}u^{\pm}$ in $\Sigma$, where $u^{\pm}$ is thescattering state at $\pm\infty$ respectively.

Some Results on the Scattering Theory for Nonlinear Schr\"{o}dinger Equations in Weighted $L^{2}$ Space