RELATIONSHIP BETWEEN SCATTERING MATRIX AND SPECTRUM OF QUANTUM GRAPHS
RELATIONSHIP BETWEEN SCATTERING MATRIX AND SPECTRUM OF QUANTUM GRAPHS
复制标题
散射矩阵与量子图谱之间的关系
DOI：
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发表时间：
2008
期刊：
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通讯作者：
Brian Winn
中科院分区：
文献类型：
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作者：
G. Berkolaiko;Brian Winn;Brian Winn
摘要我们研究了度量图上拉普拉斯算子的谱特性与相关的酉散射算子之间的等价性。我们证明了能级间距的统计数据和特征基中的观测矩在所有键长接近正常数值的限度内一致。 1 引言 近年来，量子图因其作为物理模型的适用性和有趣的数学特性而引起了广泛关注。我们建议读者参考 [1] 和即将出版的卷 [2]，以获取大量新结果。最近专门针对量子图的评论包括 [3, 4]。在本文中，我们重点关注一个有趣的特征，即使用图模型来探测量子系统的普遍性。量子力学未解决的悖论之一是，当人们在半经典体系中进行统计观察时，观察到许多量子系统非常相似。这体现在能级和相关的能量本征函数中。尽管付出了巨大的努力，但这种普遍性在数学上却很难被理解。通用量子图展示了这种普遍行为，并代表了最有可能的系统，首先会找到这种普遍性的完整数学严格证明。 [5] 中已经在这个方向上采取了重要的步骤。量子图可以用两种不同但相关的方式定义（完整的描述将出现在下一节中）。人们可以考虑拉普拉斯算子的自伴实现，或者散射矩阵方法。从数学上讲，散射方法似乎更容易处理，并且已成为大多数研究的基础[6,7,8,9,10]。此外，这些量化产生的光谱略有不同。这并不像看起来那么令人困惑，因为当在一个大的间隔上取平均值时，两个版本的光谱的统计特性被认为是一致的。本文的目的是为这一信念奠定具体的数学基础。本文的计划如下：在下一节中，我们给出描述量子图的两种方法的精确定义，并在第三节中描述我们的主要结果。第 4 节我们描述了所使用的工具，然后在第 5 节中介绍了我们结果的证明。
AbstractWe investigate the equivalence between spectral characteristics of the Laplace op-erator on a metric graph, and the associated unitary scattering operator. We provethat the statistics of level spacings, and moments of observations in the eigenbasescoincide in the limit that all bond lengths approach a positive constant value. 1 Introduction Quantum graphs have attracted much attention in recent years due both to their ap-plicability as physical models, and their interesting mathematical properties. We referthe reader to [1] and the forthcoming volume [2] for a pot-pourri of new results. Recentreviews, dedicated to quantum graphs, include [3, 4].In this article we focus on one feature of interest which is the use of graph models toprobe the the universality of quantum systems. One of the unsolved paradoxes of quantummechanics is the observation that a great many quantum systems are remarkably similarwhen one makes statistical observations in the semi-classical r´egime. This manifests itselfboth in the energy levels, and associated energy eigenfunctions. Despite a great deal ofeffort, this universality is poorly understood mathematically. Generic quantum graphsexhibit this universal behavior, and represent the most likely system for which a fullmathematically rigorous proof of this universality will first be found. Important steps inthis direction have been taken in [5].A quantum graph can be defined in two different, but related, ways (a complete de-scription appears in the following section). One may consider a self-adjoint realisationof the Laplace operator, or a scattering matrix approach. Mathematically, the scatteringapproach appears to be more tractable, and has formed the basis of most investigations[6, 7, 8, 9, 10]. Moreover, the spectra arising from these quantizations are subtly different.This is not as confusing as it might seem, since the statistical properties of both versionsof the spectrum are believed to coincide when averaged over a large interval. It is thepurpose of this article to put a concrete mathematical foundation behind this belief.The plan of the article is as follows: In the next section we give precise definitions ofthe two ways to describe quantum graphs, and in section 3 describe our main results. Insection 4 we describe the tools used and then present in section 5 the proofs of our results.