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 $i{\partial }_{t}u+\Delta u+\lambda |u{|}^{\alpha }u=0$ in $\Sigma =H^1({\mathbb{R}}^d)\cap L^2(|x{|}^2;dx)$. We show that scattering states $u^{\pm }$ exist in $\Sigma$ when ${\alpha }_d<\alpha <\frac{4}{d-2}$, $d\ge 3$, $\lambda \in \mathbb{R}$ 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 $d=1,2$), $\lambda >0$ under suitable conditions on $u_0$, 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 $\Sigma$ 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 $\Vert u_0{\Vert }_{L^2}^{\sigma }\Vert \nabla u_0{\Vert }_{L^2}<{\lambda }^{-\tau }\Vert Q{\Vert }_{L^2}^{\sigma }\Vert \nabla Q{\Vert }_{L^2}$ 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.

28 pages, no figure

References in corpus (3)

• A sharp condition for scattering of the radial 3d cubic nonlinear Schroedinger equation
• Convergence to Scattering States in the Nonlinear Schrödinger Equation
• Scattering for the focusing energy-subcritical NLS