2011/09/09 by Katsushi Ito, Erik Skibsted, Ito, K. +1
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Spectral Theory in Mathematical Physics
paper · pdf · doi:10.48550/arxiv.1109.1925
openalex publication_date 2011/09/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we introduce a notion of scattering theory for the Laplace-Beltrami operator on non-compact, connected and complete Riemannian manifolds. A principal condition is given by a certain positive lower bound of the second fundamental form of angular submanifolds at infinity. Another condition is certain bounds of derivatives up to order one of the trace of this quantity. These conditions are shown to be optimal for existence and completeness of a wave operator. Our theory does not involve prescribed asymptotic behaviour of the metric at infinity (like asymptotic Euclidean or hyperbolic metrics studied previously in the literature). A consequence of the theory is spectral theory for the Laplace-Beltrami operator including identification of the continuous spectrum and absence of singular continuous spectrum.