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 ψh = ah e i uh h , with ah(x) = e v∗ h(x)−vh(x) 2h , uh(x) = P ·x+ v ∗ h(x)+vh(x) 2 , and vh, v ∗ h satisfying the equations − h∆vh 2 + 1 2 |P +Dvh | + V = Hh(P ), h∆v∗ h 2 + 1 2 |P +Dv∗ h | + V = Hh(P ), where the constant Hh(P ) is the h effective potential and x is on the torus. L. C. Evans considered limit measures |ψh| in T, when h→ 0, for any n ≥ 1. We consider the limit measures on the phase space T×R, for n = 1, and, in addition, we obtain rigorous asymptotic expansions for the functions vh, and v ∗ h, when h→ 0. (*) Partially supported by CAMGSD/IST through FCT Program POCTI FEDER and by grants PTDC/MAT/114397/2009, UTAustin/MAT/0057/2008, PTDC/EEA-ACR/67020/2006, PTDC/MAT/69635/2006, and PTDC/MAT/72840/2006, and by the bilateral agreement Brazil-Portugal (CAPES-FCT) 248/09. (**) Partially supported by CNPq, PRONEX – Sistemas Dinâmicos, INCT, and beneficiary of CAPES financial support. (***) Partially supported by a CNPq postdoc scholarship. 1 2 DIOGO A. GOMES (*), ARTUR O. LOPES (**), AND JOANA MOHR (***)