  We construct a family of Fourier Integral Operators, defined for arbitrary
large times, representing a global parametrix for the Schr\"odinger propagator
when the potential is quadratic at infinity. This construction is based on the
geometric approach to 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.
