Geometric approach to the Hamilton-Jacobi equation and global parametrices for the Schr\"odinger propagator

TL;DR

This paper develops a geometric method to construct global parametrices for the Schr"odinger propagator with quadratic potentials, avoiding caustics and enabling semiclassical approximations.

Contribution

It introduces a geometric approach to the Hamilton-Jacobi equation that constructs Fourier Integral Operators for large times, bypassing classical flow caustics.

Findings

Constructed a family of Fourier Integral Operators for the Schr"odinger propagator.

Provided a detailed analysis of the phase function to recover WKB approximations.

Achieved a global parametrix valid for arbitrary large times with quadratic potentials.

Abstract

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.