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

Hassell, Andrew; Melrose, Richard; Vasy, András

doi:10.2140/apde.2008.1.127

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1948-206X (online)

ISSN 2157-5045 (print)

Author index

To appear

Other MSP journals

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

Andrew Hassell, Richard Melrose and András Vasy

Vol. 1 (2008), No. 2, 127–196

DOI: 10.2140/apde.2008.1.127

Abstract

In this paper, the scattering and spectral theory of $H = Δ_{g} + V$ is developed, where $Δ_{g}$ is the Laplacian with respect to a scattering metric $g$ on a compact manifold $X$ with boundary and $V \in C^{\infty} (X)$ 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 $H - σ$ 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 $σ$ is the source of solutions of $H u = σ u$ . 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 $R (σ \pm i 0)$ applied to compactly supported functions.

Keywords

radial point, scattering metric, degree zero potential, asymptotics of generalized eigenfunctions, microlocal Morse decomposition, asymptotic completeness

Mathematical Subject Classification 2000

Primary: 35P25

Secondary: 81U99

Milestones

Received: 7 January 2008

Revised: 12 September 2008

Accepted: 13 October 2008

Published: 29 July 2009

Authors

Andrew Hassell
Department of Mathematics Australian National University Canberra ACT 0200 Australia
http://wwwmaths.anu.edu.au/~hassell/index.html
Richard Melrose
Department of Mathematics Massachusetts Institute of Technology Cambridge MA 02139-4307 United States
http://www-math.mit.edu/~rbm/rbm-home.html
András Vasy
Department of Mathematics Stanford University Stanford, CA 94305-2125 United States
http://math.stanford.edu/~andras/

Vol. 1, No. 2, 2008