Non-Scattering Energies and Transmission Eigenvalues in ${\mathbb{H}}^n$

Abstract

We consider non-scattering energies and transmission eigenvalues of compactly supported potentials in the hyperbolic spaces ${\mathbb{H}}^n$. We prove that in ${\mathbb{H}}^2$ a corner bounded by two hyperbolic lines intersecting at an angle smaller than $180^{\circ }$ always scatters, and that one of the lines may be replaced by a horocycle. In higher dimensions, we obtain similar results for corners bounded by hyperbolic hyperplanes intersecting each other pairwise orthogonally, and that one of the hyperplanes may be replaced by a horosphere. The corner scattering results are contrasted by proving discreteness and existence results for the related transmission eigenvalue problems.

33 pages, 4 figures