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The goal of this paper is to present some arguments leading to the following conjecture: a formally self-adjoint differential operator on a closed manifold is essentially self-adjoint if and only if the Hamiltonian flow of its symbol is complete. This holds for differential operators of degree two on the circle, for differential operators of degree one on any closed manifold and for Lorentzian Laplacians on generic Lorentzian surfaces. RÉSUMÉ. -Le but de cet article est de présenter desdoi:10.5802/afst.1719 fatcat:rchr6n3myrgwja2a2h5yo5ihfq