量子混沌本征态的剖析
Anatomy of quantum chaotic eigenstates
摘要：
量子化混沌系统的本征函数不能用显式公式来描述，甚至不能用近似公式来描述。本综述总结了（选定的）分析方法用于描述这些本征态，在半经典极限。描述的层次是宏观的（人们希望了解光滑可观测量的量子平均值）和微观的（人们希望了解本征函数的最大值，周期轨道的“疤痕”，节点集和域的结构，局部相关性的信息），并且通常集中在统计结果上。为了理解这些统计性质，人们引入了各种“随机波函数”模型，这些模型通常与数值数据吻合得很好。我们还讨论了一些特定的系统（如算术系统），这些随机模型的出发点。
英文摘要：
The eigenfunctions of quantized chaotic systems cannot be described by explicit formulas, even approximate ones. This survey summarizes (selected) analytical approaches used to describe these eigenstates, in the semiclassical limit. The levels of description are macroscopic (one wants to understand the quantum averages of smooth observables), and microscopic (one wants informations on maxima of eigenfunctions, "scars" of periodic orbits, structure of the nodal sets and domains, local correlations), and often focusses on statistical results. Various models of "random wavefunctions" have been introduced to understand these statistical properties, with usually good agreement with the numerical data. We also discuss some specific systems (like arithmetic ones) which depart from these random models.
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