The spectrogram expansion of Wigner functions

Abstract

Wigner functions generically attain negative values and hence are not probability densities. We prove an asymptotic expansion of Wigner functions in terms of Hermite spectrograms, which are probability densities. The expansion provides exact formulas for the quantum expectations of polynomial observables. In the high frequency regime it allows to approximate quantum expectation values up to any order of accuracy in the high frequency parameter. We present a Markov Chain Monte Carlo method to sample from the new densities and illustrate our findings by numerical experiments.

Computing quantum dynamics in the semiclassical regime

“…The novel spectrogram method that combines initial sampling of positive phase space distributions with plain, uncorrected classical dynamics was proposed by Keller, Lasser and Ohsawa (2016). Higher-order spectrogram expansions have recently been analysed in Keller (2019).…”

Polyanalytic Toeplitz Operators: Isomorphisms, Symbolic Calculus and Approximation of Weyl Operators

“…As briefly discussed in the introduction, the terms Toeplitz, anti-Wick and localization operators are in parts used in interchanged ways within the literature. Classic references include [8,28], while our notation and scaling is, e.g, in accordance with [34,53].…”

“…We first recall Bargmann transforms as well as the well-known Toeplitz, Weyl and anti-Wick quantization schemes. Moreover, for the reader's convenience and later reference we recall the spectrogram expansion of Wigner functions from [34].…”

Convolutions for Berezin quantization and Berezin-Lieb inequalities

“…Remark. An example of a generalized Husimi-representation has recently been considered by Keller in [10], where σ is taken to be a finite-rank operator.…”