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

TL;DR

This paper analyzes the spectral properties of the wave equation with Dirac damping on non-compact star graphs, revealing abrupt spectral changes and instability at critical couplings, with implications for quantum mechanics.

Contribution

It introduces a novel spectral analysis of wave equations with Dirac damping on star graphs, highlighting abrupt spectral transitions and instability phenomena.

Findings

Spectral properties change abruptly at specific couplings.

The evolution problem becomes highly unstable at critical couplings.

Connections to the Dirac equation in quantum mechanics are discussed.

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.