Communications in Mathematical Physics Scarred Eigenstates for Quantum Cat Maps of Minimal Periods
Communications in Mathematical Physics Scarred Eigenstates for Quantum Cat Maps of Minimal Periods
复制标题
数学物理中的通信最小周期量子猫图的伤痕本征态
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
S. Bievre
中科院分区：
文献类型：
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作者：
F. Faure;S. Nonnenmacher;S. Bievre
本文构造了“量子阿诺德猫映射”的特征函数序列，该序列在半经典极限下，在动力学的周期轨道上表现出强烈的疤痕现象。更准确地说，这些状态有一个半经典极限测度它是环面上1/2的标准化勒贝格测度加上1/2的标准化狄拉克测度集中在任何先验给定的动力学周期轨道上。众所周知（施尼尔曼定理），“大多数”特征函数序列在环面上是等分布的。因此，我们构造的序列提供了这个一般规则的例外示例。我们的构造和证明方法利用了特殊值的存在性，其中映射的量子周期相对“短”，并且在此时间尺度上对相干态的演化进行了严格的控制。我们还提供了相空间中这些状态的点向描述，揭示了它们在固定点附近的“双曲”结构，并产生了更精确的局部化估计。
In this paper we construct a sequence of eigenfunctions of the “quantum Arnold’s cat map” that, in the semiclassical limit, shows a strong scarring phenomenon on the periodic orbits of the dynamics. More precisely, those states have a semiclassical limit measure that is the sum of 1/2 the normalized Lebesgue measure on the torus plus 1/2 the normalized Dirac measure concentrated on any a priori given periodic orbit of the dynamics. It is known (the Schnirelman theorem) that “most” sequences of eigenfunctions equidistribute on the torus. The sequences we construct therefore provide an example of an exception to this general rule. Our method of construction and proof exploits the existence of special values of for which the quantum period of the map is relatively “short”, and a sharp control on the evolution of coherent states up to this time scale. We also provide a pointwise description of these states in phase space, which uncovers their “hyperbolic” structure in the vicinity of the fixed points and yields more precise localization estimates.