In this paper, the scattering and spectral theory of $H={\Delta }_g+V$ is developed, where ${\Delta }_g$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V\in {\mathcal{C}}^{\infty }\left(X\right)$ is real; this extends our earlier results in the two-dimensional case. Included in this class of operators are perturbations of the Laplacian on Euclidean space by potentials homogeneous of degree zero near infinity. Much of the particular structure of geometric scattering theory can be traced to the occurrence of radial points for the underlying classical system. In this case the radial points correspond precisely to critical points of the restriction, $V_0$, of $V$ to $\partial X$ and under the additional assumption that $V_0$ is Morse a functional parameterization of the generalized eigenfunctions is obtained.

The main subtlety of the higher dimensional case arises from additional complexity of the radial points. A normal form near such points obtained by Guillemin and Schaeffer is extended and refined, allowing a microlocal description of the null space of $H-\sigma$ to be given for all but a finite set of “threshold” values of the energy; additional complications arise at the discrete set of “effectively resonant” energies. It is shown that each critical point at which the value of $V_0$ is less than $\sigma$ is the source of solutions of $Hu=\sigma u$. The resulting description of the generalized eigenspaces is a rather precise, distributional, formulation of asymptotic completeness. We also derive the closely related $L^2$ and time-dependent forms of asymptotic completeness, including the absence of $L^2$ channels associated with the nonminimal critical points. This phenomenon, observed by Herbst and Skibsted, can be attributed to the fact that the eigenfunctions associated to the nonminimal critical points are “large” at infinity; in particular they are too large to lie in the range of the resolvent $R\left(\sigma \pm i0\right)$ applied to compactly supported functions.

Suppose that $M$ is a strictly pseudoconvex CR manifold bounding a compact complex manifold $X$ of complex dimension $m$. Under appropriate geometric conditions on $M$, the manifold $X$ admits an approximate Kähler–Einstein metric $g$ which makes the interior of $X$ a complete Riemannian manifold. We identify certain residues of the scattering operator on $X$ as conformally covariant differential operators on $M$ and obtain the CR $Q$-curvature of $M$ from the scattering operator as well. In order to construct the Kähler–Einstein metric on $X$, we construct a global approximate solution of the complex Monge–Ampère equation on $X$, using Fefferman’s local construction for pseudoconvex domains in ${ℂ}^m$. Our results for the scattering operator on a CR-manifold are the analogue in CR-geometry of Graham and Zworski’s result on the scattering operator on a real conformal manifold.

We establish global well-posedness and scattering for solutions to the mass-critical nonlinear Schrödinger equation $i{u}_t+\Delta u=\pm |u{|}^{4∕d}u$ for large spherically symmetric $L_x^2\left({ℝ}^d\right)$ initial data in dimensions $d\ge 3$. In the focusing case we require that the mass is strictly less than that of the ground state. As a consequence, we obtain that in the focusing case, any spherically symmetric blowup solution must concentrate at least the mass of the ground state at the blowup time.