12-143 Alberto Parmeggiani - Lorenzo Zanelli

Wigner measures supported on weak KAM tori (470K, pdf) Nov 26, 12

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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.

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