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{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 $\gamma >1+\frac{4}{n}$. 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.