Quantum Graphs and Their Applications

该项目致力于量子图的一个新兴领域及其在准一维复杂物理系统建模中的应用。在这个项目中，量子图是被认为是一维奇异变种的图，它配备了微分或伪微分的“哈密顿量”。要考虑的主要问题是用图形模型来模拟薄系统中的波的传播，并研究这些模型的性质与从化学、光学到纳米技术的许多应用领域的相关性。这项研究的动机来自于工程、科学和数学的各个领域。其中包括材料科学(纳米技术，特别是含有量子线的电路，以及新型光学材料，即所谓的光子晶体)和高性能计算(光学和量子计算)。该项目旨在发展量子图论这门学科，尽管它具有广泛的适用性，但仍处于早期开发阶段。该项目将通过应用到几个活跃的科学和工程分支来促进这一重要数学领域的知识。

The project addresses an emerging field of quantum graphs and their applications to modeling quasi-one-dimensional complex physical systems. In this project, quantum graphs are graphs considered as one-dimensional singular varieties equipped with differential or pseudo-differential "Hamiltonians." The main problems to be considered are modeling wave propagation in thin systems by graph models and studying properties of these models of relevance for many areas of application, ranging from chemistry, to optics, to nanotechnology.Motivation for this study comes from various areas of engineering, sciences, and mathematics. Among them are materials science (nanotechnology, in particular circuits containing quantum wires, and novel optical materials, so-called photonic crystals) and high-performance computing (optical and quantum computing). The project aims to develop the subject of quantum graph theory, which, in spite of its wide applicability, is still in early stages of development. The project will advance knowledge in this important area of mathematics with applications to several active branches of science and engineering.