2009/06/17 by Félix Ali Mehmeti, Felix Ali Mehmeti, Robert Haller-Dintelmann +5 · 1 citation
Computer Science · Mathematics · Physics and Astronomy · #34B45 (Primary) 42A38 #47A10 #47A60 #47A70 (Secondary) #FOS: Mathematics #Nonlinear Dynamics and Pattern Formation #Nonlinear Photonic Systems #Quantum optics and atomic interactions #Spectral Theory (math.SP) #math.SP #msc:34B45 #msc:42A38 #msc:47A10 #msc:47A60 #msc:47A70
paper · pdf · doi:10.48550/arxiv.0906.3230
24 pages, 1 figure
arxiv created 2009/06/17 · openalex publication_date 2009/06/17 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We consider the Klein-Gordon equation on a star-shaped network composed of n half-axes connected at their origins. We add a potential which is constant but different on each branch. The corresponding spatial operator is self-adjoint and we state explicit expressions for its resolvent and its resolution of the identity in terms of generalized eigenfunctions. This leads to a generalized Fourier type inversion formula in terms of an expansion in generalized eigenfunctions. The characteristics of the problem are marked by the non-manifold character of the star-shaped domain. Therefore the approach via the Sturm-Liouville theory for systems is not well-suited.