Dispersive estimates and optimality for Schrödinger equations on product cones
Kouichi Taira
TL;DR
This work determines when the Schrödinger propagator on the product cone $X=C( ho\mathbb{S}^{n-1})$ exhibits the standard dispersive decay and when this decay is obstructed by geometry and the inverse-square potential. Employing a purely harmonic-analytic framework, the authors express the propagator kernel through a Gegenbauer–Bessel expansion, enabling a detailed large-$x$ asymptotic analysis via stationary and non-stationary phase arguments. Key contributions include the sharp dispersive bound $|e^{itH}|\lesssim t^{-n/2}$ for $\rho\ge 1$, refined bounds for $\rho<1$ involving the parameter $\nu_0=\sqrt{((n-2)/2)^2+c}$, and an optimality result showing failure of the dispersive estimate on certain diagonal/antipodal configurations when $\rho<1/2$ (under $\rho^{-1}\notin 2\mathbb{N}$). The results extend previous Euclidean and cone-based analyses and yield homogeneous Strichartz estimates via Keel–Tao, illustrating how the cross-section geometry controls time decay on product cones.
Abstract
In this paper, we study time decay estimates for the Schrödinger propagator on the product cone $(X,g)$, where $X=C(ρ\mathbb{S}^{n-1})=(0,\infty)\times ρ\mathbb{S}^{n-1}$. We prove that the usual dispersive estimate holds when the radius $ρ$ 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.