Nonlinear Schrödinger Equation on Real Hyperbolic Spaces

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 wellposedness results for NLS. Specifically, for small intial data, we prove L 2 and H 1 global wellposedness for any subcritical nonlinearity (in contrast with the flat case) and with no assumption of gauge invariance on the nonlinear term. On the other hand, if we assume ℑ(F (x, u))u = 0, the solution satisfies L 2 conservation of charge and hence, as in the flat case, it is possible to extend the local L 2 solutions to global ones. The corresponding argument in H 1 requires the conservation of energy, which holds under the stronger defocusing condition F = G ′ (x, |u| 2)u, G(x, s) ≥ 0. Recall that global wellposedness in the gauge invariant case was already proved by Banica, Carles and Staffilani [4], for small radial L 2 data and 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 (x, u), 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 nonlinearity : the geometry of hyperbolic spaces makes every power–like nonlinearity short range.