2010/12/14 by Félix Ali Mehmeti, Felix Ali Mehmeti, Robert Haller-Dintelmann +5
Mathematics · Physics and Astronomy · #47A10 #47A60 #47A70 #Analysis of PDEs (math.AP) #Cold Atom Physics and Bose-Einstein Condensates #FOS: Mathematics #Nonlinear Photonic Systems #Primary 34B45 #Quantum optics and atomic interactions #Secondary 42A38 #math.AP #msc:34B45 #msc:42A38 #msc:47A10 #msc:47A60 #msc:47A70
paper · pdf · doi:10.48550/arxiv.1012.3068
This article is a substantially extended version of the paper "The Klein-Gordon equation with multiple tunnel effect on a star-shaped network: Expansions in generalized eigenfunctions" (arXiv:0906.3230v1 [math.SP])
arxiv created 2010/12/14 · openalex publication_date 2010/12/14 · arxiv updated 2010/12/15 · openalex created_date 2025/10/24 · 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. Further we prove the surjectivity of the associated transformation, thus showing that it is in fact a spectral representation. 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. The considerable effort to construct explicit formulas involving the tunnel effect generalized eigenfunctions is justified for example by the perspective to study the influence of tunnel effect on the L-infinity-time decay.