  We consider the NLS with variable coefficients in dimension $n\ge3$
\begin{equation*} i \partial_t u - Lu +f(u)=0, \qquad
Lv=\nabla^{b}\cdot(a(x)\nabla^{b}v)-c(x)v, \qquad \nabla^{b}=\nabla+ib(x),
\end{equation*} on $\mathbb{R}^{n}$ or more generally on an exterior domain
with Dirichlet boundary conditions, for a gauge invariant, defocusing
nonlinearity of power type $f(u)\simeq|u|^{\gamma-1}u$. We assume that $L$ is a
small, long range perturbation of $\Delta$, plus a 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
$e^{itL}$, we prove global well posedness in the energy space for subcritical
powers $\gamma<1+\frac{4}{n-2}$, and scattering provided $\gamma>1+\frac4n$.
When the domain is $\mathbb{R}^{n}$, by extending the Strichartz estimates due
to Tataru [Tataru08], we prove that the conditional assumption is satisfied and
deduce well posedness and scattering in the energy space.
