On the dynamics of WKB wave functions whose phase are weak KAM solutions of H-J equation

In the framework of toroidal Pseudodifferential operators on the flat torus Tn := (R/2πZ)n we begin by proving the closure under composition for the class of Weyl operators Opw ~ (b) with simbols b ∈ Sm(Tn × Rn). Subsequently, we consider Opw ~ (H) when H = 1 2 |η| + V (x) where V ∈ C(T;R) and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrödinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrödinger equation to the solution of the Liouville equation on Tn ×Rn written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space H(T;C) with phase functions in the class of Lipschitz continuous weak KAM solutions (of positive and negative type) of the Hamilton-Jacobi equation 1 2 |P + ∇xv±(P, x)| 2 + V (x) = H̄(P ) for P ∈ lZn with l > 0, and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of P +∇xv±.