# Hyperbolic triangles without embedded eigenvalues

### Abstract

We consider the Neumann Laplacian acting on square-integrable functions on a triangle in the hyperbolic plane that has one cusp. We show that the generic such triangle has no eigenvalues embedded in its continuous spectrum. To prove this result we study the behavior of the real-analytic eigenvalue branches of a degenerating family of triangles. In particular, we use a careful analysis of spectral projections near the crossings of these eigenvalue branches with the eigenvalue branches of a model operator.

## Authors

Luc Hillairet

Laboiratoire MAPMO, Université d'Orléans, 45067 Orléans, France

Chris Judge

Department of Mathematics, Indiana University, Bloomington, IN