For the Schrodinger operator, the effective geometry for spectral considerations consists of the distribution of resonant wells and is controlled by the pseudo-Riemannian metric of Jacobi and Agmon. The effective geometry for the wave equation is determined by the shape of the obstacle and the boundary conditions. The effective geometry for the hyperbolic manifold is determined by the Poincare metric. In each of these cases, this research will examine how the spectral data reflects the trapping or non-trapping property of the potential or the capacity of the metric to trap geodesics. A goal of this research is the derivation of the trace formulas which express geometric properties of the manifold in terms of the spectral data. This research addresses the general question of the interplay of the geometric and spectral properties of non-compact manifolds. First, how does the geometry of the manifold determine the spectral types. Second, how does the spectral data place constraints on the geometry and topology of the manifold. Three cases of application are the quantum mechanics of ordered media with electric fields, scattering for the wave equation, and infinite volume hyperbolic manifolds.
