We consider the Schrödinger equation with no radial assumption on real hyperbolic spaces. We obtain sharp dispersive and Strichartz estimates for a large family of admissible pairs. As a first consequence, we obtain strong well-posedness results for NLS. Specifically, for small initial data, we prove L 2 and H 1 global wellposedness for any subcritical power (in contrast with the Euclidean case) and with no gauge invariance assumption on the nonlinearity F . On the other hand, if F is gauge invariant, L 2 charge is conserved and hence, as in the Euclidean case, it is possible to extend local L 2 solutions to global ones. The corresponding argument in H 1 requires conservation of energy, which holds under the stronger condition that F is defocusing. Recall that global well-posedness in the gauge invariant case was already proved by Banica, Carles and Staffilani [4], for small radial L 2 data or for large radial H 1 data. The second important application of our global Strichartz estimates is scattering for NLS both in L 2 and in H 1 , with no radial or gauge invariance assumption. Notice that, in the Euclidean case, this is only possible for the critical power γ = 1+ 4 n and can be false for subcritical powers while, on hyperbolic spaces, global existence and scattering of small L 2 solutions holds for all powers 1 < γ ≤ 1 + 4 n . If we restrict to defocusing nonlinearities F , we can extend the H 1 scattering results of [4] to the nonradial case. Also there is no distinction anymore between short range and long range nonlinearities : the geometry of hyperbolic spaces makes every power-like nonlinearity short range.

We study the dispersive properties of the wave equation associated with the shifted Laplace-Beltrami operator on real hyperbolic spaces and deduce new Strichartz estimates for a large family of admissible pairs. As an application, we obtain local well-posedness results for the nonlinear wave equation.