Inverse spectral geometry

Abstract

Determining the spectral properties of the Laplacian on a given Riemannian manifold is a “forward” spectral problem. The corresponding “inverse” problem is to deduce geometric properties from some knowledge of the spectrum. In the case of a surface with hyperbolic ends, the input data could include the resonance set, the scattering phase, perhaps even the scattering matrix for a particular set of frequencies.