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

Spectral projections and resolvent estimates on Damek-Ricci spaces and their applications