We construct a time-dependent scattering theory for Schrödinger operators
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
$R\times \partial M$, where $\partial M$ is the boundary of $M$ at infinity. We
prove the existence and the completeness of the wave operators, and show that
our scattering matrix is equivalent to the absolute scattering matrix, which is
defined in terms of the asymptotic expansion of generalized eigenfunctions. Our
method is functional analytic, and we use no microlocal analysis in this paper.