Some Results on the Scattering Theory for Nonlinear Schrödinger Equations in Weighted $L^{2}$ Space

Abstract

We investigate the scattering theory for the nonlinear Schrödinger equation $i \partial_{t}u+ Δu+λ|u|^αu=0$ in $Σ=H^{1}(\mathbb{R}^{d})\cap L^{2}(|x|^{2};dx)$. We show that scattering states $u^{\pm}$ exist in $Σ$ when $α_{d}<α<\frac{4}{d-2}$, $d\geq3$, $λ\in \mathbb{R}$ with certain smallness assumption on the initial data $u_{0}$, and when $α(d)\leq α< \frac{4}{d-2}$($α\in [α(d), \infty)$, if $d=1,2$), $λ>0$ under suitable conditions on $u_{0}$, where $α_{d}$, $α(d)$ are the positive root of the polynomial $dx^{2}+dx-4$ and $dx^{2}+(d-2)x-4$ respectively. Specially, when $λ>0$, we obtain the existence of $u^{\pm}$ in $Σ$ for $u_{0}$ below a mass-energy threshold $M[u_{0}]^σE[u_{0}]<λ^{-2τ}M[Q]^σE[Q]$ and satisfying an mass-gradient bound $\|u_{0}\|_{L^{2}}^σ\|\nabla u_{0}\|_{L^{2}}<λ^{-τ}\|Q\|_{L^{2}}^σ\|\nabla Q\|_{L^{2}}$ with $\frac{4}{d}<α<\frac{4}{d-2}$($α\in (\frac{4}{d}, \infty)$, if $d=1,2$), and also for oscillating data at critical power $α=α(d)$, where $σ=\frac{4-(d-2)α}{αd-4}$, $τ=\frac{2}{αd-4}$ and $Q$ is the ground state. We also study the convergence of $u(t)$ to the free solution $e^{itΔ}u^{\pm}$ in $Σ$, where $u^{\pm}$ is the scattering state at $\pm\infty$ respectively.