On the scattering theory for the cubic nonlinear Schrödinger and Hartree type equations in one space dimension (Q1277031)

The article is devoted to the asymptotic behaviour for large time of solutions to the Cauchy problem \[ iu_t+ {1\over 2}u_{xx} =f\bigl( | u|^2\bigr),\;u(x,0)= u_0(x)\;\bigl((x,t)\in \mathbb{R}^2\bigr) \] where either \(f(| u|^2)= \lambda\int | x-y|^{-\delta} | u|^2 (y)dy\) \((\lambda\in\mathbb{R},\;0\leq\delta<1)\), or \(f(| u|^2)=\lambda| t |^{1-\delta} | u|^2\) \((0<\delta<1)\). The initial data are such that \(e^{\beta| x|} u_0\in L^2\) (given \(\beta>0)\) with small \(L^2\)-norm. By using the Schrödinger evolution group \(U(t)=e^{it \Delta/2}\) for the linearized Cauchy problem, the authors prove the sharp decay estimate \(\| u(.,t) \|_{L^p}\leq \text{const.} t^{1/p-1/2}\) for \(t\geq 1\) and \(0\leq p<\infty\) and the existence of a unique final state \(\widehat u_+\in L^2\) satisfying \[ u(t,x)= (it)^{-1/2} \widehat u_+\left({x\over t}\right) \exp\left( {ix^2\over 2t} -{it^{1-\delta} \over 1-\delta} f\bigl(| \widehat u_+|^2 \bigr) \left({x \over t} \right)\right)+ O(t^{{1\over 2} -2\delta}) \] uniformly in \(x\), and analogous \(L^2\)-asymptotics for \(t\geq 1\).