Dispersive estimates and optimality for Schrödinger equations on product cones

In this paper, we study time decay estimates for the Schrödinger propagator on the product cone $(X,g)$, where $X=C(\rho {\mathbb{S}}^{n-1})=(0,\mathrm{\infty })\times \rho {\mathbb{S}}^{n-1}$. We prove that the usual dispersive estimate holds when the radius $\rho$ is greater than or equal to 1 and fails otherwise. A part of the former result was already established in a recent paper by Jia-Zhang. The method used here relies purely on harmonic analysis, whereas Jia-Zhang employed microlocal analysis to capture the precise asymptotic behavior of the propagator.