Spectral theory and inverse problem on asymptotically hyperbolic orbifolds

We consider an inverse problem associated with $n$-dimensional asymptotically hyperbolic orbifolds $(n \geq 2)$ having a finite number of cusps and regular ends. By observing solutions of the Helmholtz equation at the cusp, we introduce a generalized $S$-matrix, and then show that it determines the manifolds with its Riemannian metric and the orbifold structure.