Wigner measures and the semi-classical limit to the Aubry-Mather measure

TL;DR

This paper studies the semi-classical limit of Wigner measures on the tangent bundle of the 1D torus, showing their convergence to the Aubry-Mather measure and providing asymptotic expansions for related functions.

Contribution

It establishes the convergence of Wigner measures to the Mather measure for energy levels above the minimum, and derives rigorous asymptotic expansions for key functions as Planck's constant approaches zero.

Findings

Wigner measures converge to the Mather measure on the tangent bundle.

Asymptotic expansions for functions v_h and v_h^* are obtained as h approaches zero.

Results are specific to the one-dimensional torus and energy levels above the minimum.

Abstract

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 $\mu_h$ we consider are associated to $\psi_h,$ a distinguished critical solution of the Evans' quantum action given by $\psi_h=a_h\,e^{i\frac{u_h}h}$, with $a_h(x)=e^{\frac{v^*_h(x)-v_h(x)}{2h}}$, $u_h(x)=P\cdot x+\frac{v^*_h(x)+v_h(x)}{2},$ and $v_h,v^*_h$ satisfying the equations -\frac{h\, \Delta v_h}{2}+ 1/2 \, | P + D v_h \,|^2 + V &= \bar{H}_h(P), \frac{h\, \Delta v_h^*}{2}+ 1/2 \, | P + D v_h^* \,|^2 + V &= \bar{H}_h(P), where the constant $\bar{H}_h(P)$ is the $h$ effective potential and $x$ is on the torus. L.\ C.\ Evans considered limit measures $|\psi_h|^2$ in $\mathbb{T}^n$, when $h\to 0$, for any $n\geq 1$. We consider the limit measures on the phase space $\mathbb{T}^n\times\mathbb{R}^n$, for $n=1$, and, in addition, we obtain rigorous asymptotic expansions for the functions $v_h$, and $v^*_h$, when $h\to 0$.