ASYMPTOTICS FOR LARGE TIME OF SOLUTIONS TO THE NONLINEAR SCHRODINGER AND HARTREE-EQUATIONS

We study the asymptotic behavior in time of solutions to the Cauchy pr oblems for the nonlinear Schrodinger equation with a critical power no nlinearity and the Hartree equation. We prove the existence of modifie d scattering states and the sharp time decay estimate in the uniform n orm of solutions to the Cauchy problem with small initial data. This e stimate is very important for the proof of the existence of modified s cattering states to the nonlinear Schrodinger equations with a critica l nonlinearity and the Hartree equation. In order to derive the desire d estimates we introduce a certain phase function since the previous m ethods, based solely on a priori estimates of the operator x + it del acting on the solution without specifying any phase function, do not w ork for the critical case under consideration. The well-known nonexist ence of the usual L-2 scattering states shows that our result is sharp .