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.