  We construct time-dependent wave operators for Schr\"{o}dinger equations with
long-range potentials on a manifold $M$ with asymptotically conic structure. We
use the two space scattering theory formalism, and a reference operator on a
space of the form $\mathbb{R} \times \partial M$, where $\partial M$ is the
boundary of $M$ at infinity. We construct exact solutions to the
Hamilton-Jacobi equation on the reference system $\mathbb{R} \times \partial
M$, and prove the existence of the modified wave operators.
