Statistics of spectra of quantum graphs

量子图可以被认为是由连接的线组成的振动网络，可能像蜘蛛网一样连接起来，或者以更复杂的方式连接起来。在一定的振动频率下，可能会形成驻波。这些频率被称为特征值，驻波的形状被称为特征函数。值得注意的是，这些本征值和本征函数可以作为复杂量子力学系统的能级和波函数的模型，如果考虑到诸如能级之间间隔分布等统计指标的话。这种对应关系在很大程度上是一种数值观察，但也得到了启发式和严格证据的支持。本提案旨在严格理解量子图的特征函数形状的某些方面，意图利用所获得的理解来预测其他复杂量子系统。半经典量子力学的核心奥秘之一是理解波函数在大能量下的行为。已知具有混沌经典动力学的量子系统的波函数在大能量极限下几乎都是均匀分布的。这意味着，对于在这种状态下制备的量子力学粒子，在任何区域找到它的概率仅仅与该区域的体积成正比。可能的异常波函数的行为还远远没有被完全理解。一种可能性是它们根本不存在；也就是说，所有波函数都是均匀分布的。另一种可能性是，波函数在不稳定的经典周期轨道周围变得增强，这与均匀分布相距甚远。这是一个重要的问题，因为物理直觉并不能给出一个明确的预测。此外，在不同的系统中，有数学证据指向这两种结果。我相信，量子图将是这些问题的一个合适的试验场，避免了与复杂量子系统相关的技术困难。使用量子图作为模型已经在理解能级之间的间隔分布方面取得了一些成功。我希望在理解量子波函数的局域化方面取得类似的进展。这将建立在70多年来图形模型作为理解数学化学和物理光谱特征的关键工具的基础上。量子波函数中半经典结构的最新进展强调了所提出项目的及时性。

Quantum graphs may be thought of as vibrating networks of connected wires, perhaps connected like a spider's web, or else in more complicated ways. At certain frequencies of vibration, standing waves may be formed. These frequencies are called eigenvalues, and the shapes of the standing waves are called eigenfunctions. What is remarkable is that these eigenvalues and eigenfunctions serve as a model for the energy levels and wave-functions of complex quantum mechanical systems, if statistical indicators are considered such as the distribution of spacings between levels. This correspondence is largely a numerical observation, but has been supported by heuristic and rigorous evidence.The present proposal aims to understand rigorously some aspects of eigenfunction shape for quantum graphs, with the intention of using the understanding gained to make predictions about other complex quantum systems. One of the central mysteries of semi-classical quantum mechanics is to understand how wave-functions behave at large energies. It is known that almost all of the wave-functions for quantum systems which have chaotic classical dynamics become uniformly distributed in the large energy limit. This means that for a quantum mechanical particle prepared in such a state, the probability of finding it in any region is merely proportional to the volume of that region. What is yet far from being fully understood is the behaviour of possible exceptional wave-functions. One possibility is that they do not even exist; that is, all wave-functions become uniformly distributed. Another possibility is that the wave-functions become enhanced around unstable classical periodic orbits, which would be very far from uniform distribution. This is an important question, since physical intuition does not give a clear prediction one way or the other. Moreover there is mathematical evidence pointing to both outcomes, in different systems. My belief is that quantum graphs will be a suitable testing ground for these questions, avoiding the technical difficulties associated with complex quantum systems.Using quantum graphs as models has already resulted in some successes in understanding the distribution of spacings between energy levels. I hope to make similar progress in understanding localisation in quantum wave-functions. This will build upon the 70-plus year history of graph-like models as key tools in understanding spectral features of mathematical chemistry and physics. Recent progresses in semi-classical structures in quantum wave-functions emphasise the timeliness of the proposed project.