Sharp Strichartz estimates for the Schr\"odinger equation on the sphere

TL;DR

This paper establishes sharp Strichartz estimates for the Schrödinger equation on spheres, providing simple proofs based on spectral projection estimates and determining optimal regularity conditions across dimensions.

Contribution

It introduces a straightforward proof method for sharp Strichartz estimates on spheres using spectral projections, and identifies the precise regularity index for initial data in any dimension.

Findings

Sharp Strichartz estimates for the Schrödinger equation on spheres.

Simple proof techniques based on spectral projection estimates.

Optimal regularity index for initial data in spheres of dimension d ≥ 2.

Abstract

In this contribution we investigate the Schr\"ordinger equation associated to the Laplacian on the sphere in the form of sharp Strichartz estimates. We will provided simple proofs for our main theorems using purely the $L^2\rightarrow L^p$ spectral estimates for the operator norm of the spectral projections (associated to the spherical harmonics) proved in [8]. A sharp index of regularity is established for the initial data in spheres of arbitrary dimension $d\geq 2$.