Convergence to Scattering States in the Nonlinear Schr\"odinger Equation

TL;DR

This paper investigates the long-term behavior of solutions to the nonlinear Schrödinger equation, establishing conditions under which solutions converge to scattering states and analyzing the relationship between different scattering behaviors.

Contribution

It provides new results on the convergence to scattering states in the nonlinear Schrödinger equation, including conditions for equivalence of different scattering descriptions and estimates of solution norms.

Findings

Convergence to scattering states under certain conditions.

Equivalence between different scattering formulations.

Norm estimates of solutions compared to free evolution.

Abstract

In this paper, we consider global solutions of the following nonlinear Schr\"odinger equation $iu_t+\Delta u+\lambda|u|^\alpha u = 0,$ in $\R^N,$ with $\lambda\in\R,$ $\alpha\in(0,\frac{4}{N-2})$ $(\alpha\in(0,\infty)$ 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 $e^{-it\Delta}u(t)-u_ \pm\to0$ 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.