2025/10/10 by Mikuláš Kučera, Kučera, Mikuláš
Mathematics · Physics and Astronomy · #34B45 (Secondary) #35P10 (Primary) 35L05 #47D06 #47E05 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Physics (math-ph) #Spectral Theory (math.SP) #math-ph #math.AP #math.FA #math.MP #math.SP #msc:34B45 #msc:35L05 #msc:35P10 #msc:47D06 #msc:47E05
paper · pdf · doi:10.48550/arxiv.2510.09829
25 pages
arxiv created 2026/07/31 · arxiv updated 2026/08/03
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.