Taksu Cheon, Pavel Exner, Ondrej Turek
Approximation of a general singular vertex coupling in quantum graphs
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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 $\delta$ potential
and a vector potential coupled to the ``loose'' edges by a
$\delta$ 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.