Global well-posedness for Schrodinger equations with derivative

We prove that the one-dimensional Schrodinger equation with derivative in the nonlinear term is globally well-posed in H-s for s> 2/3, for small L-2 data. The result follows from an application of the "I-method." This method allows us to de ne a modi cation of the energy norm H-1 that is almost conserved and can be used to perform an iteration argument. We also remark that the same argument can be used to prove that any quintic nonlinear defocusing Schrodinger equation on the line is globally well-posed for large data in H-s, for s> 2/3.