A Mathematical Justification for the Herman-Kluk Propagator

A class of Fourier Integral Operators which converge to the unitary group of the Schroedinger equation in semiclassical limit $\text{eps}\to 0$ is constructed. The convergence is in the uniform operator norm and allows for an error bound of order $O({\text{eps}}^{1-\rho })$ 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 $\text{eps}$. In the chemical literature the approximation is known as the Herman-Kluk propagator.