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

Abstract

We investigate the scattering theory for the nonlinear Schr\"{o}dinger equation $i \partial_{t}u+ \Delta u+\lambda|u|^\alpha u=0$ in $\Sigma=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\pm}$ exist in $\Sigma$ when $\alpha_{d}<\alpha<\frac{4}{d-2}$, $d\geq3$, $\lambda\in \mathbb{R}$ with certain smallness assumption on the initial 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 the polynomial $dx^{2}+dx-4$ and $dx^{2}+(d-2)x-4$ respectively. Specially, when $\lambda>0$, we obtain the existence of $u^{\pm}$ in $\Sigma$ for $u_{0}$ 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 $d=1,2$), 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 the free solution $e^{it\Delta}u^{\pm}$ in $\Sigma$, where $u^{\pm}$ is the scattering state at $\pm\infty$ respectively.