Spectrum of the wave equation with Dirac damping on a compact star graph
Mikuláš Kučera
TL;DR
This work analyzes the damped wave equation with Dirac delta damping on a compact interval and extends the framework to compact star graphs. The authors prove that the spectrum is completely captured by zeros of the entire function $S(\lambda; a, \alpha)$, and that eigenvalue multiplicities coincide with root multiplicities, enabling precise spectral descriptions. They establish a complete Riesz-basis criterion: for rational damping placements $a=p\pi/q$, the root vectors form a Riesz basis for generic complex damping with $\alpha \neq \pm 2$, and in general, the basis property holds iff $\alpha \neq \pm 2$, with extensions to star-graph settings showing a similar pattern $\alpha \neq \pm n$ on an $n$-edge graph. The results rely on a robust combination of resolvent analysis via Green's functions, adjoint relations, and a detailed spectral-trace analysis, providing a spectral-solution pathway for damping optimization problems and clarifying the role of critical damping values in non-regular damping networks.
Abstract
We consider the wave equation with a distributional Dirac damping and Dirichlet boundary conditions on a compact interval. It is shown that the spectrum of the corresponding wave operator is fully determined by zeroes of an entire function. Consequently, a considerable change of spectral properties is shown for certain critical values of the damping parameter. We also derive a definitive criterion for the Riesz basis property of the root vectors for an arbitrary placement of a complex-valued Dirac damping. Finally, we consider a generalisation of the problem for compact star graphs and provide insight into the essence of the critical damping constant.