Pointwise dispersive estimates for Schrodinger and wave equations in a conical singular space

TL;DR

This paper establishes pointwise dispersive estimates for Schrödinger and wave equations on conical spaces, revealing a critical conjugate radius threshold of π for these estimates to hold.

Contribution

It introduces a modified Hadamard parametrix approach that bypasses conjugate point issues when the conjugate radius exceeds π, advancing understanding of dispersive behavior on conical manifolds.

Findings

Dispersive estimates hold when conjugate radius > π

Modified Hadamard parametrix is effective in this setting

Conjugate radius π is a critical threshold for L^p estimates

Abstract

We study the pointwise decay estimates for the Schr\"odinger and wave equations on a product cone $(X,g)$, where the metric $g=dr^2+r^2 h$ and $X=C(Y)=(0,\infty)\times Y$ is a product cone over the closed Riemannian manifold $(Y,h)$ with metric $h$. Under the assumption that the conjugate radius $\epsilon$ of $Y$ satisfies $\epsilon>\pi$, we prove the pointwise dispersive estimates for the Schr\"odinger and half-wave propagator in this setting. The key ingredient is the modified Hadamard parametrix on $Y$ in which the role of the conjugate points does not come to play if $\epsilon>\pi$. In a work in progress, we will further study the case that $\epsilon\leq\pi$ in which the role of conjugate points come. A new finding is that a threshold of the conjugate radius of $Y$ for $L^p$-estimates in this setting is the magical number $\pi$.