  In this paper we consider the Laplace-Beltrami operator \Delta on Damek-Ricci
spaces and derive pointwise estimates for the kernel of exp(\tau \Delta), when
\tau \in C* with Re(\tau) \geq 0. When \tau \in iR*, we obtain in particular
pointwise estimates of the Schr\"odinger kernel associated with \Delta. We then
prove Strichartz estimates for the Schr\"odinger equation, for a family of
admissible pairs which is larger than in the Euclidean case. This extends the
results obtained by Anker and Pierfelice on real hyperbolic spaces. As a
further application, we study the dispersive properties of the Schr\"odinger
equation associated with a distinguished Laplacian on Damek-Ricci spaces,
showing that in this case the standard dispersive estimate fails while suitable
weighted Strichartz estimates hold.
