Convergence to Scattering States in the Nonlinear Schrödinger Equation

Abstract

In this paper, we consider global solutions of the following nonlinear Schrödinger equation $i{u}_t+\Delta u+\lambda |u{|}^{\alpha }u=0,$ in ${\mathbb{R}}^N,$ with $\lambda \in \mathbb{R},$ $\alpha \in (0,\frac{4}{N-2})$ $(\alpha \in (0,\infty )$ if $N=1)$ and \linebreak $u(0)\in X\equiv H^1({\mathbb{R}}^N)\cap L^2(|x{|}^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-it\Delta }u(t)-u_{\pm }\to 0$ in $X$ as $t\to \pm \infty$ then $u(t)-e^{it\Delta }{u}_{\pm }\to 0$ in $X$ as $t\to \pm \infty .$ We also study the converse. Finally, we estimate $|\Vert u(t){\Vert }_X-\Vert e^{it\Delta }{u}_{\pm }{\Vert }_X|$ under some less restrictive assumptions.