Mather Measures Associated with a Class of Bloch Wave Functions

In this paper we study the Wigner transform for a class of smooth Bloch wave functions on the flat torus ${\mathbb{T}^n = \mathbb{R}^n /2\pi \mathbb{Z}^n}$ : $$\psi_{\hbar,P} (x) = a (\hbar,P,x) {\rm e}^{ \frac{i}{\hbar} ( P\cdot x + \hat{v}(\hbar,P,x) )}.$$ On requiring that ${P \in \mathbb{Z}^n}$ and ${\hbar = 1/N}$ with ${N \in \mathbb{N}}$ , we select amplitudes and phase functions through a variational approach in the quantum states space based on a semiclassical version of the classical effective Hamiltonian ${{\bar H}(P)}$ which is the central object of the weak KAM theory. Our main result is that the semiclassical limit of the Wigner transform of ${\psi_{\hbar,P}}$ admits subsequences converging in the weak* sense to Mather probability measures on the phase space. These measures are invariant for the classical dynamics and Action minimizing.