Scattering in the energy space for the NLS with variable coefficients

Abstract

We consider the NLS with variable coefficients in dimension $$n\ge 3$$ $$\begin{aligned} i \partial _t u - Lu +f(u)=0, \qquad Lv=\nabla ^{b}\cdot (a(x)\nabla ^{b}v)-c(x)v, \qquad \nabla ^{b}=\nabla +ib(x), \end{aligned}$$ 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 $$\gamma >1+\frac{4}{n}$$ . When the domain is $$\mathbb {R}^{n}$$ , 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.