Pointwise dispersive estimates for Schr\"odinger operators on product cones

TL;DR

This paper establishes dispersive estimates for Schr"odinger equations on product cones with radial potentials, revealing dimension-dependent decay rates and spectral measure effects, extending Euclidean results to cone geometries.

Contribution

It provides the first detailed dispersive estimates for Schr"odinger operators on product cones with radial potentials, accounting for geometric and spectral complexities.

Findings

In odd dimensions, decay rates match Euclidean space results.

In even dimensions, decay is slower by a factor of t^{1/2}.

Weighted L^1 to L^∞ estimates are established for solutions.

Abstract

We investigate the dispersive properties of solutions to the Schr\"odinger equation with a weakly decaying radial potential on cones. If the potential has sufficient polynomial decay at infinity, then we show that the Schr\"odinger flow on each eigenspace of the link manifold satisfies a weighted $L^1\to L^\infty$ dispersive estimate. In odd dimensions, the decay rate we compute is consistent with that of the Schr\"odinger equation in a Euclidean space of the same dimension, but the spatial weights reflect the more complicated regularity issues in frequency that we face in the form of the spectral measure. In even dimensions, we prove a similar estimate, but with a loss of $t^{1/2}$ compared to the sharp Euclidean estimate.