Resolvent and spectral measure for Schrödinger operators on flat Euclidean cones

Abstract

We construct the Schwartz kernel of resolvent and spectral measure for Schrödinger 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, ${\mathbb{S}}_{\sigma }^1=\mathbb{R}/2\pi \sigma \mathbb{Z}$, with radius $\sigma >0$ and the metric $g=d{r}^2+r^{2}d{\theta }^2$. As products, we prove the dispersive estimates for the Schrödinger and half-wave propagators in this setting.

13 pages; Comments are welcome!