  We show that in the presence of a rough external potential, a 3d cubic
defocusing NLS is globally well-posed in H^s for s>5/6. The proof is based on
the approach of Colliander-Keel-Staffilani-Takaoka-Tao, called the I-method,
but in order to deal with a rough potential, we modify harmonic analysis tools
by spectral theory.
