Remarks on the Fundamental Solution to Schrödinger Equation with Variable Coefficients

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We consider Schrödinger operators H on Rⁿ with variable coefficients. Let H₀=-12 be the free Schrödinger operator and we suppose H is a "short-range" perturbation of H₀. 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-itH₀ 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.