Asymptotics for large time of solutions to the nonlinear Schrödinger and Hartree equations

We study the asymptotic behavior in time of solutions to the Cauchy problems for the nonlinear Schrödinger equation with a critical power nonlinearity and the Hartree equation. We prove the existence of modified scattering states and the sharp time decay estimate in the uniform norm of solutions to the Cauchy problem with small initial data. This estimate is very important for the proof of the existence of modified scattering states to the nonlinear Schrödinger equations with a critical nonlinearity and the Hartree equation. In order to derive the desired estimates we introduce a certain phase function since the previous methods, based solely on a priori estimates of the operator x + it∇ acting on the solution without specifying any phase function, do not work for the critical case under consideration. The well-known nonexistence of the usual L² scattering states shows that our result is sharp.