Spectrum of the wave equation with Dirac damping on a non-compact star graph

TL;DR

This work analyzes the damped wave equation on non-compact star graphs with damping modeled by a complex Robin vertex condition. By studying the non-self-adjoint generator $\mathcal{W}_\alpha$, the authors reveal a sharp spectral transition: for generic $\alpha$ the spectrum lies on $i\mathbb{R}$, but at the critical couplings $\alpha=\pm N$ (where $N$ is the number of edges) the spectrum fills a full complex half-plane, driving instability in the evolution. They establish norm-resolvent convergence of regularized dampings to the Dirac-type damping as a limit model, and they connect these damping phenomena to Dirac-type operators in relativistic quantum mechanics. The results illuminate how vertex-localized, non-self-adjoint damping can drastically alter stability and spectral structure on networks, with implications for quantum graphs and wave control on extended geometries.

Abstract

We consider the wave equation on non-compact star graphs, subject to a distributional damping defined through a Robin-type vertex condition with complex coupling. It is shown that the non-self-adjoint generator of the evolution problem admits an abrupt change in its spectral properties for a special coupling related to the number of graph edges. As an application, we show that the evolution problem is highly unstable for the critical couplings. The relationship with the Dirac equation in non-relativistic quantum mechanics is also mentioned.

Theorems & Definitions (32)

• Theorem 2.1
• Theorem 2.2
• Theorem 2.3
• Theorem 2.4
• Proposition 3.1
• Proposition 3.2
• proof
• Lemma 3.3
• proof
• Proposition 3.4