arXiv 1111.31872011-11-14math.DSmath-phmath.MPmath.PRquant-ph

Wigner测度与Aubry-Mather测度的半经典极限

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

Diogo A. Gomes, Artur O. Lopes, Joana Mohr

摘要：

本文研究了一维环面切丛上Wigner测度的半经典极限的渐近性质。特别是，我们表明收敛的维格纳措施的马瑟措施的切丛，以上的有效哈密顿量的最小值的能级。我们考虑的Wigner测度$μ_h$与由$ψ_h=a_h\,e^{i\frac{u_h}h}$给出的Evans量子作用量的一个特殊临界解$ψ_h,$相关联，其中$a_h(x)=e^{\frac{v^*_h(x)-v_h(x)}{2h}}$，$u_h(x)=P\cdot x+\frac{v^*_h(x)+v_h(x)}{2},$和$v_h,v^*_h$满足方程- \frac{h\, Δv_h}{2} + 1/2 \，|P + D v_h \，|^2 + V &= \bar{H}_h（P），\frac{h\, Δv_h^*}{2} + 1/2 \，|P + D v_h^* \，|其中常数$\bar{H}_h(P)$是$h$的有效电势，$x$位于环面上。L. C. Evans考虑了$\mathbb{T}^n$中的极限措施$|ψ_h|^2$，当$h\to 0$时，对于任何$n\geq 1$。我们认为相空间上的极限措施$\mathbb{T}^n\times\mathbb{R}^n$，$n=1$，此外，我们得到严格的渐近展开的功能$v_h$，$v^*_h$，当$h\to 0$。

英文摘要：

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 $μ_h$ we consider are associated to $ψ_h,$ a distinguished critical solution of the Evans' quantum action given by $ψ_h=a_h\,e^{i\frac{u_h}h}$, 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\, Δv_h}{2}+ 1/2 \, | P + D v_h \,|^2 + V &= \bar{H}_h(P), \frac{h\, Δv_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 $|ψ_h|^2$ 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 $h\to 0$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

摘要：

英文摘要：

相关论文