The Geometry of the Semiclassical Wave Front Set for Schrödinger Eigenfunctions on the Torus

This paper deals with the phase space analysis for a family of Schr¨odinger eigenfunctions ψ on the flat torus by the semiclassicalWave Front Set. We study those ψ such that WF is contained in the graph of the gradient of some viscosity solutions of the Hamilton-Jacobi equation. It turns out that the semiclassical Wave Front Set of such Schr¨odinger eigenfunctions is stable under viscous perturbations of Mean Field Game kind. These results provide a further viewpoint, and in a wider setting, of the link between the smooth invariant tori of Liouville integrable Hamiltonian systems and the semiclassical localization of Schr¨odinger eigenfunctions on the torus.