Alberto Parmeggiani - Lorenzo Zanelli
Wigner measures supported on weak KAM tori
(470K, pdf)

ABSTRACT.  In the setting of the Weyl quantization of a classical system described by an arbitrary $C^\infty$ Tonelli Hamiltonian $H$ on $\mathbb{T}^n 	imes \mathbb{R}^n$ 
(i.e. $\eta \mapsto H(x,\eta)$ is strictly convex and uniformly superlinear for every $x \in \mathbb{T}^n$) we exhibit a class of wave functions with 
uniquely associated Wigner probability measures. In particular, we obtain measures that are invariant under the Hamiltonian dynamics and with support contained in weak KAM tori in the phase space. 
These sets are the graphs of weak (Lipschitz) KAM solutions of the stationary Hamilton-Jacobi equation. 
Morever, we show that such Wigner measures are in fact given by the Legendre transform of Mather probability measures, which are are characterized to be invariant under the 
Lagrangian dynamics, and Action minimizing.