Approximation of a general singular vertex coupling in quantum graphs

Taksu Cheon Pavel Exner Ondej Turek

TBD mathscidoc:1910.43046

Annals of Physics, 325, (3), 548-578, 2010.3
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 potential and a vector potential coupled to the loose edges by a 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.
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@inproceedings{taksu2010approximation,
  title={Approximation of a general singular vertex coupling in quantum graphs},
  author={Taksu Cheon, Pavel Exner, and Ondej Turek},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020122935786707575},
  booktitle={Annals of Physics},
  volume={325},
  number={3},
  pages={548-578},
  year={2010},
}
Taksu Cheon, Pavel Exner, and Ondej Turek. Approximation of a general singular vertex coupling in quantum graphs. 2010. Vol. 325. In Annals of Physics. pp.548-578. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191020122935786707575.
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