Linear restriction estimates for Schroedinger equation on metric cones

TL;DR

This paper establishes modified linear restriction estimates for the Schrödinger equation on metric cones, leading to global Strichartz estimates and scattering results for small initial data in specific geometric settings.

Contribution

It introduces new linear restriction estimates for Schrödinger operators on metric cones with angular regularity, enabling advanced analysis of solutions.

Findings

Global-in-time Strichartz estimates for radial initial data

Small data scattering theory for mass-critical nonlinear Schrödinger equation

Modified restriction estimates tailored to metric cone geometry

Abstract

In this paper, we study some modified linear restriction estimates of the dynamics generated by Schroedinger operator on metric cone $M$, where the metric cone $M$ is of the form $M=(0,\infty)_r\times\Sigma$ with the cross section $\Sigma$ being a compact $(n-1)$-dimensional Riemannian manifold $(\Sigma,h)$ and the equipped metric is $g=\mathrm{d}r^2+r^2h$. Assuming the initial data possesses additional regularity in angular variable $\theta\in\Sigma$, we show some linear restriction estimates for the solutions. As applications, we obtain global-in-time Strichartz estimates for radial initial data and show small initial data scattering theory for the mass-critical nonlinear Schroedinger equation on two-dimensional metric cones.