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The Van Vleck Formula on Ehrenfest Time Scales and Stationary Phase Asymptotics for Frequency-Dependent Phases

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Abstract

The Van Vleck formula is a semiclassical approximation to the integral kernel of the propagator associated to a time-dependent Schrödinger 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 \(|t| \le c|\log \hbar |\). 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 (Rev Math Phys 22(10):1123-1145, 2010) , 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.

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Notes

  1. This is a indeed a rough statement since we do not make any formal hypotheses on \({\hat{H}}\) at this stage. In particular, it is significant to have a hypothesis to ensure that \(\Xi _t (x,y)\) is a finite set (cf. Remark 6 following Theorem 1.2). Our results below will introduce a microlocal cutoff in momentum to achieve this.

  2. Here we treat the Maslov index in a manner consistent with [16], allowing it to take on a half-integer value. This means the expression here differs slightly from that in some other treatments, where the first factor is \((2\pi i\hbar )^{-\frac{d}{2}}\) instead of \((2\pi \hbar )^{-\frac{d}{2}}\) as we have here. This convention also agrees with the one in [21], where half-integer indices naturally appear for paths starting from the Maslov cycle.

  3. Not all of these works address (1.5) explicitly, but they are all at least small variations on it. For this reason, we do not attempt to compare the results themselves, and instead emphasize the methods behind them.

  4. The approximation seems closely related to those resulting from a wave packet transform such as the Bargmann or FBI transforms. But in the works cited here, the derivation of the amplitude a does not make significant use of such tools. Instead, (1.9) is viewed as an ansatz, and integration by parts is used to inductively determine \(a_0,a_1,a_2,\dots .\) Nonetheless, such transforms were used in [20] to obtain estimates on the error in the approximations.

  5. We state the \(\lambda <\frac{1}{4}\) hypothesis from [20] here, but in order to apply Theorem 1.1 towards the Van Vleck formula we use the stronger hypothesis \(24\lambda + 6 \delta < 1\) below.

  6. Here we display the effect of including all terms in the asymptotic expansion \(H \sim \sum _{j=0}^{\infty } \hbar ^j H_j\), whereas this is treated implicitly in [20, §3].

  7. Here and below, the roles of \({\tilde{a}}\) and \({\tilde{a}}_j\) will be much different than their role in Remark 3.5.

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Acknowledgements

The author is grateful to Didier Robert and Peter Miller for helpful comments on this work. He was supported in part by the National Science Foundation Grant DMS-1565436.

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  1. Department of Mathematics and Statistics, University of New Mexico, Albuquerque, NM, USA

    Matthew D. Blair

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Blair, M.D. The Van Vleck Formula on Ehrenfest Time Scales and Stationary Phase Asymptotics for Frequency-Dependent Phases. Commun. Math. Phys. 392, 517–543 (2022). https://doi.org/10.1007/s00220-022-04384-z

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