A result is announced concerning a family of semiclassical Fourier Integral Operators representing a global parametrix for the Schrödinger propagator when the potential is quadratic at infinity. The construction is based on the geometrical approach of the corresponding Hamilton-Jacobi equation and thus sidesteps the problem of the caustics generated by the classical flow. Moreover, a detailed study of the real phase function allows us to recover a WKB semiclassical approximation which necessarily involves the multivaluedness of the graph of the Hamiltonian flow past the caustics.
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Sandro Graffi, Lorenzo Zanelli, Global parametrices for the Schrödinger propagator and geometric approach to the Hamilton-Jacobi equation. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 22 (2011), no. 1, pp. 17–28DOI 10.4171/RLM/585