A Mathematical Justification for the Herman-Kluk Propagator

TL;DR

This paper provides a rigorous mathematical foundation for the Herman-Kluk propagator, demonstrating its convergence to the Schrödinger evolution in the semiclassical limit with explicit error bounds.

Contribution

It constructs a class of Fourier Integral Operators that approximate the Schrödinger group with proven convergence and error estimates, clarifying the mathematical basis of the Herman-Kluk propagator.

Findings

Convergence in the uniform operator norm as ps to the Schrf6dinger group.

Error bounds of order O(ps^{1- ho}) for Ehrenfest timescales.

Improved error estimates for shorter times of order O(1).

Abstract

A class of Fourier Integral Operators which converge to the unitary group of the Schroedinger equation in semiclassical limit $\eps\to 0$ is constructed. The convergence is in the uniform operator norm and allows for an error bound of order $O(\eps^{1-\rho})$ for Ehrenfest timescales, where $\rho$ can be made arbitrary small. For the shorter times of order O(1), the error can be improved to arbitrary order in $\eps$. In the chemical literature the approximation is known as the Herman-Kluk propagator.