Bégout, Pascal (2001) Convergence to Scattering States in the Nonlinear Schrödinger Equation. Communications in Contemporary Mathematics (ccm), 3 (3). pp. 403-418.

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Abstract

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