Scattering in the energy space for the NLS with variable coefficients

We consider the NLS with variable coefficients in dimension 3 or larger on an exterior domain with Dirichlet boundary conditions, and with a gauge invariant, defocusing nonlinearity of power type growing of order p. We assume that the principal part of the operator is a small, long range perturbation of the Laplacian, plus a small magnetic potential, plus an electric potential with a large positive part. The first main result of the paper is a bilinear smoothing (interaction Morawetz) estimate for the solution. As an application, under the conditional assumption that Strichartz estimates are valid for the linear flow, we prove global well posedness in the energy space for subcritical powers, and scattering provided the power is larger than 1+4/n. Further, when the domain is Rn, by extending the Strichartz estimates due to Tataru, we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space.