Microlocal propagation near radial points and scattering for symbolic potentials of order zero

Abstract

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.