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. On requiring the Planck constant h to be of the form 1/N, for some positive integer N, we select amplitudes and phase functions through a variational approach in the quantum-state space based on a semiclassical version of the classical effective Hamiltonian which is the central object of the weak KAM theory. Our main result is that the semiclassical limit of the Wigner transform of the considered Bloch wave functions 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.