Convergence to Scattering States in the Nonlinear Schrödinger Equation

Abstract

In this paper, we consider global solutions of the following nonlinear Schrödinger equation $iu_t+Δu+λ|u|^αu = 0,$ in $\R^N,$ with $λ\in\R,$ $α\in(0,\frac{4}{N-2})$ $(α\in(0,\infty)$ if $N=1)$ and \linebreak $u(0)\in X\equiv H^1(\R^N)\cap L^2(|x|^2;dx).$ We show that, under suitable conditions, if the solution $u$ satisfies $e^{-itΔ}u(t)-u_ \pm\to0$ in $X$ as $t\to\pm\infty$ then $u(t)-e^{itΔ}u_\pm\to0$ in $X$ as $t\to\pm\infty.$ We also study the converse. Finally, we estimate $|\:\|u(t)\|_X-\|e^{itΔ}u_\pm\|_X\:|$ under some less restrictive assumptions.