CONVERGENCE TO SCATTERING STATES IN THE NONLINEAR SCHRÖDINGER EQUATION

Abstract

In this paper, we consider global solutions of the following nonlinear Schrödinger equation iut+Δ u+λ|u|αu=0, in ℝN, with λ∈ℝ, [Formula: see text] (α∈ (0,∞) if N=1) and u(0)∈X≡ H1(ℝN)∩ L2(|x|2;dx). We show that, under suitable conditions, if the solution u satisfies e-itΔu(t)-u±→0 in X as t→±∞ then u(t)-eitΔu±→0 in X as t→±∞. We also study the converse. Finally, we estimate |‖u(t)‖X- ‖eitΔu±‖X| under some less restrictive assumptions.