Summary: We consider Schrödinger operators \(H\) on \(\mathbb R^n\) with variable coefficients. Let \(H_0=-\frac12\Delta\) be the free Schrödinger operator and we suppose \(H\) is a “short-range” perturbation of \(H_0\). Then, under the nontrapping condition, we show that 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 A. Hassell and J. Wunsch [in: Partial differential equations and inverse problems. Proceedings of the Pan-American Advanced Studies Institute on partial differential equations, nonlinear analysis and inverse problems, Santiago, Chile, January 6–18, 2003. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 362, 199–209 (2004; Zbl 1062.35098)], 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.