On completeness of modified wave operators for defocusing NLS

In this manuscript, we study modified scattering for the nonlinear defocusing Schr\"odinger equation with a critical gauge-invariant nonlinearity of order 1+2/n. We address the following question: Given initial data in an appropriate weighted Sobolev space, what is the leading term in the asymptotic behavior of the solution as times goes to infinity? More precisely, we seek a final state in a space of type similar to the space of the initial data such that the leading term is represented by the free propagator and modified phase function. The solution to this problem can be reformulated in terms of the completeness of wave operators. For n=1, we obtain a complete answer, provided appropriate control on the sup - norm even for large initial data. For n = 2 completeness is established under suitable control of the sup - norm of the solution.