Mathematical Research Letters

Volume 26 (2019)

Number 2

Locating resonances on hyperbolic cones

Pages: 365 – 381



Dean Baskin (Department of Mathematics, Texas A&M University, College Station, Tx., U.S.A.)

Jeremy L. Marzuola (Department of Mathematics, University of North Carolina, Chapel Hill, N.C., U.S.A.)


In this note we explicitly compute the resonances on hyperbolic cones. These are manifolds diffeomorphic to $\mathbb{R}_{+} \times Y$ and equipped with the singular Riemannian metric $dr^2 + \mathrm{sinh}^2 r h$, where $Y$ is a compact manifold without boundary and h is a Riemannian metric on $Y$. The calculation is based on separation of variables and Kummer’s connection formulae for hypergeometric functions. To our knowledge this is the one of the few explicit resonance calculations that does not rely on the resolvent being a two-point function.

The first author was supported in part by U.S. NSF Grant DMS–1500646. The second author was supported in part by U.S. NSF Grants DMS–1312874 and DMS-1352353.

Received 10 February 2017

Accepted 13 February 2018

Published 12 August 2019