Quantum Graphs and Their Applications

该项目致力于量子图的新兴领域及其在模拟准一维复杂物理系统中的应用。在这个项目中，量子图被认为是配备有微分或伪微分“哈密顿算子”的一维奇异变种。“要考虑的主要问题是通过图形模型对薄系统中的波传播进行建模，并研究这些模型与许多应用领域（从化学到光学，再到纳米技术）相关的特性。这项研究的动机来自工程，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，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形理论的主题，尽管其广泛的适用性，但仍处于发展的早期阶段。 该项目将推进数学这一重要领域的知识，并将其应用于科学和工程的几个活跃分支。

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.