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

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.