Schrödinger equation on non-compact symmetric spaces

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