Global well-posedness for Schrödinger equations with derivative

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We prove that the 1D Schrödinger equation with derivative in the nonlinear term is globally well-posed in Hˢ, for s>2/3 for small L² data. The result follows from an application of the ``I-method''. This method allows to define a modification of the energy norm H¹ 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 Schrödinger equation on the line is globally well-posed for large data in Hˢ, for s>2/3.