Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schrödinger propagator

Details Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schrödinger propagator Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schrödinger propagator

2010-06-09

arxiv.org/abs/1006.1753v1

Physics

Mathematical Physics

Scientific paper

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.

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