Pure and Applied Mathematics Quarterly

Volume 19 (2023)

Number 4

Special Issue in honor of Victor Guillemin

Guest Editors: Yael Karshon, Richard Melrose, Gunther Uhlmann, and Alejandro Uribe

Classical and Quantum mechanics on 3D contact manifolds

Pages: 1839 – 1852

DOI: https://dx.doi.org/10.4310/PAMQ.2023.v19.n4.a5


Yves Colin de Verdière (Institut Fourier, Université Grenoble-Alpes, Saint Martin d’Hères, France)


In this survey paper, I describe some aspects of the dynamics and the spectral theory of sub-Riemannian 3D contact manifolds. We use Toeplitz quantization of the characteristic cone as introduced by Louis Boutet de Monvel and Victor Guillemin. We also discuss trace formulae following our work as well as the Duistermaat–Guillemin trace formula.

Dedicated to Victor, teacher and friend

Received 9 September 2021

Received revised 23 July 2022

Accepted 3 October 2022

Published 20 November 2023