# Scattering resonances on truncated cones

## TL;DR

This paper investigates the distribution of scattering resonances for the Laplacian on truncated Riemannian cones, providing explicit asymptotic formulas and extending previous results on non-truncated cones.

## Contribution

It constructs the resolvent and scattering matrix for truncated cones and derives explicit asymptotic distribution of resonances, extending prior work on non-truncated cones.

## Key findings

- Resonances on truncated cones are asymptotically distributed as A*r^n + o(r^n).
- Laplacian on non-truncated cones has no resonances away from zero.
- Explicit coefficient A in the resonance distribution formula.

## Abstract

We consider the problem of finding the resonances of the Laplacian on truncated Riemannian cones. In a similar fashion to Cheeger--Taylor, we construct the resolvent and scattering matrix for the Laplacian on cones and truncated cones. Following Stefanov, we show that the resonances on the truncated cone are distributed asymptotically as Ar^n + o(r^n), where A is an explicit coefficient. We also conclude that the Laplacian on a non-truncated cone has no resonances away from zero.