  We prove $L^p-L^{p^\prime}$ boundedness of spectral projections and the
resolvent of the Laplace-Beltrami operator on Damek-Ricci spaces with the
explicit norms in terms of the spectral parameter. To prove these results we
established pointwise sharp bounds on the spherical functions and their
derivatives. As an application, we study the eigenvalue bounds of Schr\"odinger
operators with complex valued potential.
