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 μ_h we consider are associated to ψ_h, a distinguished critical solution of the Evans' quantum action given by ψ_h=a_h e^{ifrac{u_h}h}, with a_h(x)=e^{frac{v^*_h(x)-v_h(x)}{2h}}, u_h(x)=Pcdot x+frac{v^*_h(x)+v_h(x)}{2}, and v_h,v^*_h satisfying the equations -\frac{h\, Δv_h}{2}+ 1/2 \, | P + D v_h \,|^2 + V &= \bar{H}_h(P), \frac{h\, Δv_h^*}{2}+ 1/2 \, | P + D v_h^* \,|^2 + V &= \bar{H}_h(P), where the constant bar{H}_h(P) is the h effective potential and x is on the torus. L.\ C.\ Evans considered limit measures |ψ_h|² in mathbb{T}ⁿ, when h→ 0, for any ngeq 1. We consider the limit measures on the phase space mathbb{T}^ntimesmathbb{R}ⁿ, for n=1, and, in addition, we obtain rigorous asymptotic expansions for the functions v_h, and v^*_h, when h→ 0.