Strichartz estimates for the Schr\"odinger equation for the sublaplacian on complex spheres

TL;DR

This paper establishes Strichartz estimates with fractional derivative loss for the Schr"odinger equation on complex spheres, leading to local well-posedness results for solutions with spectral localization.

Contribution

It introduces Strichartz estimates for the Schr"odinger equation on complex spheres with the sublaplacian, including fractional derivative loss, and derives well-posedness results based on spectral properties.

Findings

Proved Strichartz estimates with fractional loss of derivatives.

Derived local well-posedness results for the nonlinear Schr"odinger equation.

Achieved stronger results for solutions with spectral data in a proper cone.

Abstract

We study the nonlinear Schr\"odinger equation associated with the sublaplacian L on the unit sphere $S^{2n+1}$ in $C^{n+1}$ equipped with its natural CR structure. We first prove Strichartz estimates with fractional loss of derivatives for the solutions of the free Schr\"odinger equation and we then deduce some local in time well-posedness results. Our results are stated in terms of certain Sobolev-type spaces, that measure the regularity of functions on $S^2n+1$ differently according to their spectral localization. Stronger conclusions are obtained for particular classes of solutions, corresponding to initial data whose spectrum is contained in a proper cone of $N^2$.