Nontrapping surfaces of revolution with long living resonances

TL;DR

This paper investigates surfaces of revolution formed by modifying a cone with a hyperbolic cusp, revealing that some nontrapping geometries can still have infinitely many long-living resonances, challenging previous assumptions.

Contribution

It demonstrates the existence of nontrapping surfaces of revolution that possess infinitely many long-living resonances, a novel finding in geometric scattering theory.

Findings

Nontrapping surfaces can have infinitely many long-living resonances.

Surfaces with hyperbolic cusps exhibit unique resonance properties.

Long-living resonances occur despite nontrapping geodesic flow.

Abstract

We study resonances of surfaces of revolution obtained by removing a disk from a cone and attaching a hyperbolic cusp in its place. These surfaces include ones with nontrapping geodesic flow (every maximally extended non-reflected geodesic is unbounded) and yet infinitely many long living resonances (resonances with uniformly bounded imaginary part, i.e. decay rate).