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arXiv 2202.08609nlin.CDmath-phmath.MP
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混沌系统中$S$矩阵能量关联和时滞的半经典方法

Semiclassical approach to $S$ matrix energy correlations and time delay in chaotic systems

Marcel Novaes

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6 pages
发表信息：
Phys. Rev. E 105, 044213 (2022)
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摘要：

在混沌系统中，连接入射波和出射波的$M$维散射矩阵$S(E)$总是幺正的，但对能量有复杂的依赖关系。这部分地编码在由$S(E+ε)S^\dagger(E-ε)$的幂的迹线构造的、在$E$上平均的、以及由时间延迟算子$Q(E)=-i\hbar S^\dagger dS/dE$的统计特性构造的迭代器中。利用时间反演对称性破缺系统的半经典方法，我们导出了两类能量耗散子的表达式：一类是以$ε$的有理函数为系数的幂级数形式，另一类是以$ε$的有理函数为系数的幂级数形式.从后者，我们提取一个明确的公式$\rm{Tr}(Q^n)$，这是有效的所有$n$，并与随机矩阵理论的预测。

英文摘要：

The $M$-dimensional scattering matrix $S(E)$ which connects incoming to outgoing waves in a chaotic systyem is always unitary, but shows complicated dependence on the energy. This is partly encoded in correlators constructed from traces of powers of $S(E+ε)S^\dagger(E-ε)$, averaged over $E$, and by the statistical properties of the time delay operator, $Q(E)=-i\hbar S^\dagger dS/dE$. Using a semiclassical approach for systems with broken time reversal symmetry, we derive two kind of expressions for the energy correlators: one as a power series in $1/M$ whose coefficients are rational functions of $ε$, and another as a power series in $ε$ whose coefficients are rational functions of $M$. From the latter we extract an explicit formula for $\rm{Tr}(Q^n)$ which is valid for all $n$ and is in agreement with random matrix theory predictions.

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