The Laplace operator on a hyperbolic manifold I. Spectral and scattering theory

Abstract

Using techniques of stationary scattering theory for the Schrödinger equation, we show absence of singular spectrum and obtain incoming and outgoing spectral representations for the Laplace-Beltrami operator on manifolds M n arising as the quotient of hyperbolic n-dimensional space by a geometrically finite, discrete group of hyperbolic isometries. We consider manifolds M n of infinite volume. In subsequent papers, we will use the techniques developed here to analytically continue Eisenstein series for a large class of discrete groups, including some groups with parabolic elements.