Zineb Zellak: Metric Graphs with Herglotz-Nevanlinna Vertex Interactions

Abstract: Quantum graphs have emerged as a powerful framework for modeling wave propagation on networklike structures. In this work, we construct a class of operators acting as second-order differential operators along the edges of finite compact metric graphs, coupled with finite-dimensional components at the vertices, leading to vertex interactions described by Herglotz–Nevanlinna functions. This interaction mechanism naturally gives rise to spectral parameter–dependent vertex conditions, extending classical coupling models. The operator is first shown to be symmetric, and its self-adjointness is established through a detailed analysis of the resolvent equation. Finally, we investigate the spectrum of the operator in the case of a lasso graph and describe its main spectral properties.