Schr\"odinger equation on noncompact symmetric spaces

TL;DR

This paper derives sharp kernel and dispersive estimates for the Schr"odinger equation on non-compact symmetric spaces, leading to improved global well-posedness and scattering results compared to Euclidean space.

Contribution

It establishes new sharp-in-time estimates and global Strichartz inequalities for Schr"odinger equations on non-compact symmetric spaces, leveraging geometric and harmonic analysis techniques.

Findings

Sharp-in-time kernel estimates achieved

Global Strichartz inequalities established for larger admissible pairs

Global well-posedness and scattering results obtained for lower regularity data

Abstract

We establish sharp-in-time kernel and dispersive estimates for the Schr\"odinger equation on non-compact Riemannian symmetric spaces of any rank. Due to the particular geometry at infinity and the Kunze-Stein phenomenon, these properties are more pronounced in large time and enable us to prove the global-in-time Strichartz inequality for a larger family of admissible couples than in the Euclidean case. Consequently, we obtain the global well-posedness for the corresponding semilinear equation with lower regularity data and some scattering properties for small powers which are known to fail in the Euclidean setting. The crucial kernel estimates are achieved by combining the stationary phase method based on a subtle barycentric decomposition, a subordination formula of the Schr\"odinger group to the wave propagator and an improved Hadamard parametrix.