Wigner measures and the semi-classical limit to the aubry-mather measure

In this paper, we investigate the asymptotic behavior of the semi-classical limit of Wigner measures defined on the tangent bundle of the one-dimensional torus. In particular, we show the convergence of Wigner measures to the Mather measure on the tangent bundle, for energy levels above the minimum of the effective Hamiltonian.The Wigner measures μh we consider are associated to ψh, a distinguished critical solution of the Evans' quantum action given by, with, and vh,v*h satisfying the equations where the constant is the h effective potential and x is on the torus. Evans considered limit measures |ψh| ² in, when h→0, for any n≥1.We consider the limit measures on the phase space for n=1, and, in addition, we obtain rigorous asymptotic expansions for the functions vh, and v*h, when h→0.