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

2010/06/09 by Sandro Graffi, Graffi, Sandro, Lorenzo Zanelli +1
Mathematics · Physics and Astronomy · #Advanced Mathematical Physics Problems #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #math-ph #math.MP

paper · pdf · doi:10.48550/arxiv.1006.1753

arxiv created 2010/06/09 · openalex publication_date 2010/06/09 · arxiv updated 2010/06/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We construct a family of Fourier Integral Operators, defined for arbitrary large times, representing a global parametrix for the Schrödinger 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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