Randomized final-data problem for Systems of Nonlinear Schr\"odinger Equations and the Gross-Pitaevskii Equation

We consider the final-data problem for systems of nonlinear Schr"odinger equations with $L^2$ subcritical nonlinearity. An asymptotically free solution is uniquely obtained for almost every randomized asymptotic profile in $L^2({\mathbb{R}}^d)$, extending the result of J. Murphy to powers equal to or lower than the Strauss exponent. In particular, systems with quadratic nonlinearity can be treated in three space dimensions, and by the same argument, the Gross-Pitaevskii equation in the energy space. The extension is by use of the Strichartz estimate with a time weight.