  The purpose of this paper is to describe certain CR-covariant differential
operators on a strictly pseudoconvex CR manifold $M$ as residues of the
scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold
$X$ having $M$ as a `CR-infinity.' We also characterize the CR $Q$-curvature in
terms of the scattering operator. Our results parallel earlier results of
Graham and Zworski \cite{GZ:2003}, who showed that if $X$ is an asymptotically
hyperbolic manifold carrying a Poincar\'{e}-Einstein metric, the $Q$-curvature
and certain conformally covariant differential operators on the `conformal
infinity' $M$ of $X$ can be recovered from the scattering operator on $X$. The
results in this paper were announced in \cite{HPT:2006}.
