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Approximation of a general singular vertex coupling in quantum graphs |
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| Authors: | Taksu Cheon Pavel Exner Ond?ej Turek |
| Affiliation: | a Laboratory of Physics, Kochi University of Technology, Tosa Yamada, Kochi 782-8502, Japan b Doppler Institute for Mathematical Physics and Applied Mathematics, Czech Technical University, B?ehová 7, 11519 Prague, Czech Republic c Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 25068 ?e? near Prague, Czech Republic d Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Trojanova 13, 12000 Prague, Czech Republic |
| Abstract: | The longstanding open problem of approximating all singular vertex couplings in a quantum graph is solved. We present a construction in which the edges are decoupled; an each pair of their endpoints is joined by an edge carrying a δ potential and a vector potential coupled to the “loose” edges by a δ coupling. It is shown that if the lengths of the connecting edges shrink to zero and the potentials are properly scaled, the limit can yield any prescribed singular vertex coupling, and moreover, that such an approximation converges in the norm-resolvent sense. |
| Keywords: | Quantum graphs Boundary conditions Singular vertex coupling Quantum wires |
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