Spectral Properties of Quantum Graphs and Related Systems

该项目解决了与作用于度量图上定义的函数空间的微分算子谱有关的一些相关问题。具有这样一个算子的图通常被称为“量子图”。本研究有两个中心主题：研究量子图的特征值之间的相关性，以及理解它们的特征函数的形态。对于一般的图族，特征值表现为与大型随机矩阵的特征值相关联。对于非泛型图，期望有不同的相关性。本项目将从数学和物理的角度增进对这些相关性如何产生的理解。所得的结果将有助于理解数学物理中出现的其他微分算子的特征值中相关性的外观。该项目的第二个主题是理解量子图的特征函数的统计性质。对于一般图，期望特征函数表现出类似于在其他系统中已经证明存在的等分布性质。该项目的中心目标是为量子图的特征函数制定一个类似的理论。也有人怀疑稀疏的特征函数序列可能成为局部的（伤痕累累的）而不是等分布的。该项目的另一部分将是发现这种情况是否会发生，以及在什么条件下可能会发生。量子图是一个新的数学分支的一部分，有时被称为“纳米数学”，它的范围包括纳米结构中的输运、量子信息和超导性等领域。在这种情况下，我们对量子图的研究是理解量子力学理论的一部分，因为它与经典牛顿动力学（所谓的“量子混沌”）有关。基于波动的量子力学理论在当前微芯片制造的数量级尺度上是不可避免的。复杂量子力学系统中出现的波的形状和频率可以通过观察网络上振荡的类似问题来理解。该项目的结果将与当前许多不同领域的介观物理和工程研究相关。为了吸引新的研究人员进入该领域，该项目的一部分将作为本科生研究经验计划的主题，并将编写介绍性的研究生文本。

The project addresses a number of related questions concerning the spectra of differential operators acting on spaces of functions defined on metric graphs. A graph equipped with such an operator is generally referred to as a "quantum graph." The research has two central themes: investigating the correlations between the eigenvalues of quantum graphs, and understanding the morphology of their eigenfunctions. For generic families of graphs the eigenvalues appear to be correlated like eigenvalues of large random matrices. For non-generic graphs, different correlations are expected. The present project will improve the understanding of how these correlations arise both from a mathematical and physical point of view. The results obtained will be useful in understanding the appearance of correlations in the eigenvalues of other differential operators arising in mathematical physics. The second theme of the project is to understand the statistical properties of eigenfunctions of quantum graphs. For generic graphs, it is expected that the eigenfunctions should exhibit some equidistribution properties similar to those already proved to exist in other systems. A central objective of the project is to formulate an analogous theory for eigenfunctions of quantum graphs. It is also suspected that a sparse sequence of eigenfunctions may become localized (scarred) rather than equidistributed. Another part of the project will be to discover whether this can, in fact, occur, and the conditions under which it may do so. brbrQuantum graphs are part of a new branch of mathematics, sometimes called ``nanomathematics'', which contains within its scope areas such as transport in nanostructures, quantum information, and superconductivity. In this context, our study of quantum graphs is part of an effort to understand the theory of quantum mechanics as it relates to classical Newtonian dynamics (so-called "quantum chaology"). The wave-based theory of quantum mechanics is unavoidable on scales of the order of magnitude of present microchip manufacturing. The shapes and frequencies of waves appearing in complex quantum mechanical systems can be understood by looking at similar questions for oscillations on networks. The results of the project will be relevant to current research in many diverse areas of mesoscopic physics and engineering. In order to draw new researchers into the field, part of the project will be used as topics for an undergraduate research experience program, and an introductory graduate text will be written.