Existence of wave operators with time-dependent modifiers for the Sch\"odinger equations with long-range potentials on scattering manifolds

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 $\mathbb{R}\times \mathrm{\partial }M$, where $\mathrm{\partial }M$ 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.