The fundamental solution and Strichartz estimates for the Schr\"odinger equation on flat euclidean cones

TL;DR

This paper derives the fundamental solution and establishes Strichartz estimates for the Schr"odinger equation on flat Euclidean cones, analyzing asymptotic behaviors near the cone point and at infinity.

Contribution

It provides new asymptotic expansions of the fundamental solution on Euclidean cones and proves Strichartz estimates in this geometric setting.

Findings

Asymptotic expansions valid near the cone point and at infinity.

Uniform asymptotics across geometric and diffractive fronts.

Strichartz estimates established for Schr"odinger propagator on cones.

Abstract

We study the Schr\"odinger equation on a flat euclidean cone $\mathbb{R}_+ \times \mathbb{S}^1_\rho$ of cross-sectional radius $\rho > 0$, developing asymptotics for the fundamental solution both in the regime near the cone point and at radial infinity. These asymptotic expansions remain uniform while approaching the intersection of the "geometric front", the part of the solution coming from formal application of the method of images, and the "diffractive front" emerging from the cone tip. As an application, we prove Strichartz estimates for the Schr\"odinger propagator on this class of cones.