It is known that geometric optics can be derived as the high-frequency limit of the wave equation, from both experimental and theoretical perspectives. This fact can be regarded as an instance of a "quantum-classical correspondence principle", made rigorous by Egorov's theorem.

 

We will present a version of Egorov's theorem for the (half-)wave equation with a Laplace--Beltrami operator on the Euclidean space. This result allows to describe the propagation of energy at different scales, given by a family of so-called Weyl-Hörmander metrics on phase space.

 

As an application, we shall recover the so-called Poisson relation on Riemannian tori (a statement which relates the spectrum of the Laplace--Beltrami operator and the lengths of closed geodesics), giving an alternative proof of works of Chazarain and Duistermaat--Guillemin.