Scattering resonances on truncated cones

Abstract

We consider the problem of finding the resonances of the Laplacian on truncated Riemannian cones. In a similar fashion to Cheeger and 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 $A{r}^n+o\left(r^n\right)$, where $A$ is an explicit coefficient. We also conclude that the Laplacian on a nontruncated cone has no resonances.