Convergence to Scattering States in the Nonlinear Schrödinger Equation

In this paper, we consider global solutions of the following nonlinear Schrödinger equation iut + ∆u + λ|u| αu = 0, in RN, with λ ∈ R, α ∈ (0, 4 N−2) (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−it∆u(t) − u ± → 0 in X as t → ± ∞ then u(t) − e it∆u ± → 0 in X as t → ±∞. We also study the converse. Finally, we estimate | ‖u(t)‖X − ‖e it∆u±‖X | under some less restrictive assumptions