论文标题
具有分段恒定电势和多模式图的度量图的痕量公式
A trace formula for metric graphs with piecewise constant potentials and multi-mode graphs
论文作者
论文摘要
我们将量子图的散射方法推广到具有分段恒定电势和多种激发模式的量子图。免费的单模外壳是众所周知的，并导致了罗斯，科特斯和斯米兰斯基的痕量公式。通过引入有效的减少散射图片，我们可以在更通用的环境中引入新的精确痕量公式。后者是详细得出和讨论的，其中一些数字示例用于说明。我们的概括是由实验应用和基本理论考虑的动机。自由的单模量子图是现实的极端理想化，由于模型的简单性，该图可以理解大量的通用或通用现象。我们通过考虑仅在阈值高于阈值能量上的逃生模式的影响来提高一些理想化。总体而言，理论上如何在封闭的模型中做到这一点是一个具有挑战性的基本理论兴趣问题，我们在这里为量子图实现了这一点。
We generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth, Kottos and Smilansky. By introducing an effective reduced scattering picture we are able to introduce new exact trace formulas in the more general setting. The latter are derived and discussed in details with some numerical examples for illustration. Our generalization is motivated by both experimental applications and fundamental theoretical considerations. The free single-mode quantum graphs are an extreme idealization of reality that, due to the simplicity of the model allows to understand a large number of generic or universal phenomena. We lift some of this idealization by considering the influence of evanescent modes that only open above threshold energies. How to do this theoretically in a closed model in general is a challenging question of fundamental theoretical interest and we achieve this here for quantum graphs.
