Annales Academiæ Scientiarum Fennicæ
Volumen 45, 2020, 547-576


Emilia Blåsten and Esa V. Vesalainen

University of Helsinki, Department of Mathematics and Statistics
P.O. Box 64 (Pietari Kalmin katu 5), 00014 Helsingin yliopisto, Finland; emilia.blasten 'at'

Åbo Akademi University, Mathematics and Statistics
Domkyrkotorget 1, 20500 Åbo, Finland; esa.vesalainen 'at'

Abstract. We consider non-scattering energies and transmission eigenvalues of compactly supported potentials in the hyperbolic spaces Hn. We prove that in H2 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.

2010 Mathematics Subject Classification: Primary 35P25, 58J50, 35R30, 51M10, 58J05.

Key words: Hyperbolic geometry, interior transmission problem, corner, non-scattering.

Reference to this article: E. Blåsten and E. V. Vesalainen: Non-scattering energies and transmission eigenvalues in Hn. Ann. Acad. Sci. Fenn. Math. 45 (2020), 547-576.

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