Locating resonances on hyperbolic cones

TL;DR

This paper explicitly computes the resonances on hyperbolic cones, a class of hyperbolic manifolds with conic singularities, using separation of variables and hypergeometric functions, providing a rare explicit calculation method.

Contribution

It presents one of the few explicit resonance calculations on hyperbolic cones that does not depend on the resolvent as a two-point function.

Findings

Explicit formulas for resonances on hyperbolic cones

Use of hypergeometric functions and separation of variables

Advances understanding of spectral properties of hyperbolic manifolds with singularities

Abstract

In this note we explicitly compute the resonances on hyperbolic cones. These are hyperbolic manifolds with a conic singularity equipped with a warped product metric. 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 calculations of resonances that does not rely on the resolvent being a two-point function.