Resolvent and spectral measure for Schr\"odinger operators on flat Euclidean cones

Abstract

We construct the Schwartz kernel of resolvent and spectral measure for Schr\"odinger operators on the flat Euclidean cone $(X,g)$, where $X=C(\mathbb{S}_\sigma^1)=(0,\infty)\times \mathbb{S}_\sigma^1$ is a product cone over the circle, $\mathbb{S}_\sigma^1=\R/2\pi \sigma\Z$, with radius $\sigma>0$ and the metric $g=dr^2+r^2 d\theta^2$. As products, we prove the dispersive estimates for the Schr\"odinger and half-wave propagators in this setting.