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

TL;DR

This paper establishes scattering results for the nonlinear Schrödinger equation in weighted $L^2$ space under various conditions on the nonlinearity power, initial data, and parameters, extending understanding of long-term behavior of solutions.

Contribution

It provides new existence results for scattering states in weighted $L^2$ space for a range of nonlinear powers and initial data conditions, including near critical and oscillating cases.

Findings

Existence of scattering states in weighted $L^2$ space for certain nonlinear powers.

Conditions under which solutions converge to free solutions at infinity.

Results applicable to both subcritical and critical nonlinearities.

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.