On the Dynamics of WKB Wave Functions Whose Phase are Weak KAM Solutions of H-J Equation

摘要

In the framework of toroidal Pseudodifferential operators on the flat torus we begin by proving the closure under composition for the class of Weyl operators with symbols . Subsequently, we consider when where and we exhibit the toroidal version of the equation for the Wigner transform of the solution of the Schrodinger equation. Moreover, we prove the convergence (in a weak sense) of the Wigner transform of the solution of the Schrodinger equation to the solution of the Liouville equation on written in the measure sense. These results are applied to the study of some WKB type wave functions in the Sobolev space with phase functions in the class of Lipschitz continuous weak KAM solutions (positive and negative type) of the Hamilton-Jacobi equation for with , and to the study of the backward and forward time propagation of the related Wigner measures supported on the graph of .