Scattering Theory for Conformally Compact Metrics with Variable Curvature at Infinity

✍ Scribed by David Borthwick

• Publisher: Elsevier Science
• Year: 2001
• Tongue: English
• Weight: 445 KB
• Volume: 184
• Category: Article
• ISSN: 0022-1236
• DOI: 10.1006/jfan.2001.3770

✦ Synopsis

We develop the scattering theory of a general conformally compact metric by treating the Laplacian as a degenerate elliptic operator (with non-constant indicial roots) on a compact manifold with boundary. Variability of the roots implies that the resolvent admits only a partial meromorphic continuation, and the bulk of the paper is devoted to studying the structure of the resolvent, Poisson, and scattering kernels for frequencies outside the region of meromorphy. For low frequencies the scattering matrix is shown to be a pseudodifferential operator with frequency dependent domain. In particular, generalized eigenfunctions exhibit L 2 decay in directions where the asymptotic curvature is sufficiently negative. We explicitly construct the resolvent kernel for generic frequency in this part of the continuous spectrum.