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Non-scattering energies and transmission eigenvalues in \(H^n\). (English) Zbl 1446.35087

Summary: 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 \(180 °\) 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.

MSC:

35P25 Scattering theory for PDEs
58J50 Spectral problems; spectral geometry; scattering theory on manifolds
35R30 Inverse problems for PDEs
51M10 Hyperbolic and elliptic geometries (general) and generalizations
58J05 Elliptic equations on manifolds, general theory

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