Schr\"odinger equation on locally symmetric spaces

TL;DR

This paper establishes dispersive and Strichartz estimates for Schr"odinger equations on certain locally symmetric spaces, with applications to nonlinear equations' well-posedness and scattering.

Contribution

It provides the first dispersive and Strichartz estimates for Schr"odinger equations on these spaces, specifically for rank one or complex G cases.

Findings

Dispersive estimates are proven for the Schr"odinger equation on these spaces.

Strichartz estimates are established, enabling analysis of nonlinear problems.

Applications include well-posedness and scattering results for nonlinear Schr"odinger equations.

Abstract

We prove dispersive and Strichartz estimates for Schr\"o- dinger equations on a class of locally symmetric spaces {\Gamma}\X, where X = G/K is a symmetric space and {\Gamma} is a torsion free discrete sub- group of G. We deal with the cases when either X has rank one or G is complex. We present Strichartz estimates applications to the well-posedness and scattering for nonlinear Schr\"odinger equations.