  In this article we study some aspects of dispersive and concentration
phenomena for the Schr\"odinger equation posed on hyperbolic space
$\mathbb{H}^n$, in order to see if the negative curvature of the manifold gets
the dynamics more stable than in the Euclidean case. It is indeed the case for
the dispersive properties : we prove that the dispersion inequality is valid,
in a stronger form than the one on $\mathbb{R}^n$. However, the geometry does
not have enough of an effect to avoid the concentration phenomena and the
picture is actually worse than expected. The critical nonlinearity power for
blow-up turns out to be the same as in the euclidean case, and we prove that
there are more explosive solutions for critical and supercritical
nonlinearities.
