Conic singularities, generalized scattering matrix, and inverse scattering on asymptotically hyperbolic surfaces

TL;DR

This paper studies inverse problems on asymptotically hyperbolic surfaces with conical singularities, showing that a generalized scattering matrix uniquely determines the surface's metric and singularity structure.

Contribution

It introduces a generalized S-matrix for asymptotically hyperbolic surfaces with conical singularities and proves its uniqueness in determining the surface's geometry.

Findings

The generalized S-matrix encodes the geometric and singularity data.

The inverse problem is solvable for a class of non-compact hyperbolic surfaces.

The method extends scattering theory to surfaces with conical singularities.

Abstract

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with a Fuchsian group of the 1st kind $\Gamma$ containing parabolic elements. $\mathcal M$ is then non-compact, and has a finite number of cusps and elliptic singular points, which is regarded as a hyperbolic orbifold. We introduce a class of Riemannian surfaces with conical singularities on its finite part, having cusps and regular ends at infinity, whose metric is asymptotically hyperbolic. By observing solutions of the Helmholtz equation at the cusp, we define a generalized S-matrix. We then show that this generalized S-matrix determines the Riemannian metric and the structure of conical singularities.