The Van Vleck Formula on Ehrenfest time scales and stationary phase asymptotics for frequency-dependent phases

TL;DR

This paper rigorously derives the Van Vleck semiclassical approximation for quantum propagators on Ehrenfest time scales using stationary phase asymptotics for frequency-dependent phases, based on the Herman-Kluk approximation.

Contribution

It provides a rigorous derivation of the Van Vleck formula on Ehrenfest time scales utilizing stationary phase methods for frequency-dependent phases, extending previous approximations.

Findings

Validates Van Vleck approximation on Ehrenfest time scales

Develops stationary phase asymptotics for frequency-dependent phases

Connects Herman-Kluk approximation to Van Vleck formula

Abstract

The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schr\"odinger equation. Under suitable hypotheses, we present a rigorous treatment of this approximation which is valid on "Ehrenfest time scales", i.e. $\hbar$-dependent time intervals which most commonly take the form $|t| \leq c|\log\hbar|$. Our derivation is based on an approximation to the integral kernel often called the "Herman-Kluk approximation", which realizes the kernel as an integral superposition of Gaussians parameterized by points in phase space. As was shown by Robert, this yields effective approximations over Ehrenfest time intervals. In order to derive the Van Vleck approximation from the Herman-Kluk approximation, we are led to develop stationary phase asymptotics where the phase functions depend on the frequency parameter in a nontrivial way, a result which may be of independent interest.