  We consider Schr\"odinger operators $H$ on $R^n$ with variable coefficients.
Let $H_0=-\frac12\triangle$ be the free Schr\"odinger operator and we suppose
$H$ is a "short-range" perturbation of $H_0$. Then, under the nontrapping
condition, we show the time evolution operator: $e^{-itH}$ can be written as a
product of the free evolution operator $e^{-itH_0}$ and a Fourier integral
operator $W(t)$, which is associated to the canonical relation given by the
classical mechanical scattering. We also prove a similar result for the wave
operators. These results are analogous to results by Hassell and Wunsch, but
the assumptions, the proof and the formulation of results are considerably
different. The proof employs an Egorov-type theorem similar to those used in
previous works by the authors combined with a Beals-type characterization of
Fourier integral operators.
