Scattering in the energy space for the NLS with variable coefficients

We consider the NLS with variable coefficients in dimension n≥ 3 $$egin{aligned} i partial _t u - Lu +f(u)=0, qquad Lv= abla ^{b}cdot (a(x) abla ^{b}v)-c(x)v, qquad abla ^{b}= abla +ib(x), end{aligned}$$i∂tu-Lu+f(u)=0,Lv=∇b·(a(x)∇bv)-c(x)v,∇b=∇+ib(x),on Rn or more generally on an exterior domain with Dirichlet boundary conditions, for a gauge invariant, defocusing nonlinearity of power type f(u) ≃ | u| γ - 1u. We assume that L is a small, long range perturbation of Δ , 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 ei t L, we prove global well posedness in the energy space for subcritical powers γ<1+4n-2, and scattering provided γ>1+4n. When the domain is Rn, by extending the Strichartz estimates due to Tataru (Am J Math 130(3):571–634, 2008), we prove that the conditional assumption is satisfied and deduce well posedness and scattering in the energy space