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

We investigate the scattering theory for the nonlinear Schr"{o}dinger equation $i{\mathrm{\partial }}_{t}u+\mathrm{\Delta }u+\lambda \mathrm{\mid }u{\mathrm{\mid }}^{\alpha }u=0$ in $\mathrm{\Sigma }=H^1({\mathbb{R}}^d)\cap L^2(\mathrm{\mid }x{\mathrm{\mid }}^2;dx)$. We show that scattering states $u^{\pm }$ exist in $\mathrm{\Sigma }$ when ${\alpha }_d<\alpha <\frac{4}{d-2}$, $d\ge 3$, $\lambda \in \mathbb{R}$ with certain smallness assumption on the initial data $u_0$, and when $\alpha (d)\le \alpha <\frac{4}{d-2}$($\alpha\in [\alpha(d), \infty)$,if$d=1,2$),$\lambda>0$undersuitableconditionson$u_{0}$,where$\alpha_{d}$,$\alpha(d)$ are the positive root of the polynomial $d{x}^2+dx-4$ and $d{x}^2+(d-2)x-4$ respectively. Specially, when $\lambda >0$, we obtain the existence of $u^{\pm }$ in $\mathrm{\Sigma }$ for $u_0$ below a mass-energy threshold $M[u_0{]}^{\sigma }E[u_0]<{\lambda }^{-2\tau }M[Q{]}^{\sigma }E[Q]$ and satisfying an mass-gradient bound $|u_{0}|{L^{2}}^{\sigma}|\nabla u{0}|{L^{2}}<\lambda^{-\tau}|Q|{L^{2}}^{\sigma}|\nabla Q|_{L^{2}}$with$\frac{4}{d}<\alpha<\frac{4}{d-2}$($\alpha\in (\frac{4}{d}, \infty)$,if$d=1,2$),andalsoforoscillatingdataatcriticalpower$\alpha=\alpha(d)$, 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 the free solution $e^{it\mathrm{\Delta }}{u}^{\pm }$ in $\mathrm{\Sigma }$, where $u^{\pm }$ is the scattering state at $\pm \mathrm{\infty }$ respectively.