Análise de Ressonâncias em Grafos Quânticos

Abstract

Since the emergence of graph theory, attributed to Euler in the 18th century, solving problems modeled by graphs has motivated researchers across various fields. In quantum physics, two notable applications are graph states and quantum graphs. The latter has attracted attention due to its role in scattering studies from both theoretical and experimental perspectives. By studying scattering on graphs, we can observe how a particle interacts with a potential or a barrier, which can be interpreted, in this context, as the vertices of the graph equipped with specific boundary conditions. This analysis allows us to identify the emergence of resonances. To determine them, it is necessary to compute the resonance condition. Several methods can be employed for this purpose, such as the scattering matrix, the resolvent, and the Green’s function. These methodologies enable the identification of the poles of the graph, which can be classified as resonances based on their positions in the complex plane. In this work, we provide a brief review of these methods and show their equivalence. Furthermore, we analyze the behavior of the transmission amplitude probability and the poles of certain graphs as some parameters are varied.