On the scattering theory for the cubic nonlinear schrödinger and hartree type equations in one space dimension

We study the scattering problem and asymptotics for large time of solutions to the Cauchy problem for the nonlinear schrödinger and Hartree type equations wit h subcritical nonlinearities [Formula] where the nonlinear interaction term is F(|u|²) = V * |u|², V(x) = λ|x|^{− δ}, λ ∈ R, 0 < δ < 1 in the Hartree type case, or F(|u|²) = λ|t|^{1 −δ}|u|² in the case of the cubic nonlinear Schrödinger equation. We suppose that the initial data e^β|x|u₀ ∈ L², β > 0 with sufficiently small norm ε = ||e^β|x|u₀||L². Then we prove the sharp decay estimate ||u(t)||L^p ≤ C ϵt 1/p − 1/2, for all t ≥ 1 and for every 2 ≤ p ≤ ∞. Furthermore we show that for 1/2 < δ < 1 there exists a unique final state û + ∈ L² such that for all t ≥ 1 [Formula] and uniformly with respect to x [formula] where ϕ ^ denotes the Fourier transform of ϕ. Our results show that the regularity condition on the initial data which was assumed in the previous paper [9] is not needed. Also a smoothing effect for the solutions in an analytic function space is discussed.